Earlier results and the higher-degree consequence

The power-free-value problem has a long history. Ricci obtained the asymptotic in the range \(k\ge d\) (Ricci 1933). For \(d\ge3\), Erdős proved infinitude for exponent \(d-1\), and Hooley obtained the corresponding asymptotic (Erdős 1953; Hooley 1967); see also the historical discussion in (Heath-Brown 2013, Introduction). The cubic squarefree case therefore belongs to this classical range. Already in 1953, Erdős singled out the unresolved squarefreeness of \(n^4+2\) (Erdős 1953, 425). In his 1965 survey he returned to this example when describing the obstacle at exponent \(d-2\) (Erdős 1965, sec. 6, p. 219).

Nair obtained the asymptotic for \(k\ge(\sqrt2-\tfrac12)d\) (Nair 1976), with further developments by Nair and Huxley–Nair (Nair 1979; Huxley and Nair 1980). The 1976 bound reaches \(k=d-2\) for \(d\ge24\). Heath-Brown’s affine determinant method gave the range \(k\ge(3d+2)/4\) (Heath-Brown 2006, Theorem 16), reaching \(k=d-2\) for \(d\ge10\). Combining that method with Salberger’s global determinant estimates, Browning improved the range to \(k\ge(3d+1)/4\) (Browning 2011), which reaches \(k=d-2\) for \(d\ge9\). We use the formulation and published reproof in Xiao (Xiao 2017, Theorem 9.1 and Section 9): the input interval is \(1\le n\le X\) and the constant is the Euler product over all residue classes modulo \(p^k\).

The shape of the polynomial can permit stronger estimates. Heath-Brown proved an asymptotic with a power-saving error for irreducible binomials \(x^d+c\) when \(k\ge(5d+3)/9\) (Heath-Brown 2013, Theorem 1). At \(k=d-2\) this covers such binomials in degrees \(d\ge6\), while imposing the special binomial form. Reuss obtained a power-saving error for the general degree-\(d\) polynomial at exponent \(d-1\) (Reuss 2015, Theorem 2). The quartic theorem above concerns squarefree values at positive integer arguments for each fixed admissible polynomial.

Other important directions involve different hypotheses or quantifiers. Granville showed that the \(abc\) conjecture yields the predicted squarefree-value density for separable integer polynomials with no fixed prime-square divisor (Granville 1998, Theorem 1). Greaves and Helfgott studied squarefree values of binary forms; Helfgott’s Theorems 5.1–5.2 distinguish univariate cubics from homogeneous forms in two variables of degree at most six (Greaves 1992; Helfgott 2004). A density in two integer variables does not by itself give the density on the fixed line with second coordinate one. Recent results of Browning–Shparlinski and Sofos study the discrepancy from the predicted squarefree count on average over polynomial coefficient families (Browning and Shparlinski 2024, Theorems 1.1–1.2), (Sofos 2026, Theorem 1.1); those averaged statements do not imply a theorem for every fixed polynomial.

Carella’s preprint claims squarefree-value asymptotics for both \(n^4+1\) and \(n^4+2\) (Carella 2023, Theorems 1.1–1.2). Zapata Ceballos–Jalalvand claim positive density for monic irreducible degree-\(2q\) polynomials with squarefree fixed divisor whose root field contains a degree-\(q\) Galois subfield, where \(q\) is prime; this includes \(n^4+1\) (Zapata Ceballos and Jalalvand 2026, Theorem 1.1). We do not use either claim as a theorem input. Our large-prime estimate and sieve transfer are proved independently below.

The new argument in this paper concerns degrees four through eight. For higher degrees, Browning’s theorem supplies a direct consequence.

The factorwise large-prime estimates also give a fixed-product consequence. Here a nonzero polynomial is separable over \(\mathbb Q\) if each of its nonconstant irreducible factors occurs with multiplicity one.

The first assertion requires the value of the entire product \(F\) to be \(k\)-power-free. The second imposes power-freeness separately on the \(F_i(n)\), not on their product, and uses the joint local condition \(\Omega_p\) rather than a product of individual local factors. All polynomials, their number, and \(k\) are fixed; no uniform or power-saving error is asserted here. The proof is at the end of Section 2.

Our method combines number-field factorization with approximating determinants, following the framework developed by Reuss for power-free polynomial values (Reuss 2015, secs. 3–6). Heath-Brown’s determinant arguments for squarefree values of \(n^2+1\) and for power-free binomial values provide related precedents (Heath-Brown 2012, 2013). The real determinant method originates in work of Bombieri–Pila; Heath-Brown developed the prime-adic and approximating forms used in this line of arguments, and Salberger developed their local Hilbert and global congruence aspects (Bombieri and Pila 1989; Heath-Brown 2002, 2009; Salberger 2007, 2023). The uniform affine counting theorem, the explicit weighted Hilbert calculations and the quartic surface and curve arguments needed here are proved below.

The large-prime problem

The finite-prime part of the problem follows from the Chinese remainder theorem. For primes outside a fixed finite set, separability and Hensel lifting give \(\rho_f(p^k)\le d\). Consequently, for fixed large \(Y\), primes \(Y<p\le X\) remove at most \[O_f\bigl(XY^{1-k}+\pi(X)\bigr)\] integers in \((X,2X]\). The remaining task is to control primes larger than the length of the input interval. We prove the following estimate.

The proposition does not require local admissibility. In Section 2, we prove explicitly that it implies Theorem 1, including the stated sign and content generality. The density argument first holds the finite-prime cutoff fixed while \(X\) tends to infinity, and then lets the cutoff tend to infinity. This order gives the exact Euler product without an assertion of a power-saving error in the density asymptotic.

Proof strategy

Fix a dyadic prime interval \(P\le p\le2P\), and choose one such prime for each input being counted. Write \(P=X^\eta\) and \(b=d-k\eta\). The elementary relation \(f(n)=yp^k\) puts the chosen triples on the surface \(f(x)=yz^k\), with coordinate sizes \(X,X^b,X^\eta\). A weighted determinant argument on this surface suffices when \(\eta\) is large. The harder range requires more information from the factorization of \(f\).

Let \(\theta\) be a root of \(f\) and let \(K=\mathbb Q(\theta)\). Ideal factorization and the unit theorem produce integral elements \(\alpha,\beta\in K\) satisfying \[\alpha^k\beta=\mu(n-\theta)\] for a fixed nonzero integer \(\mu\). Their conjugates are balanced: the sizes of \(\alpha\) and \(\beta\) at each embedding are respectively comparable to \(P^{1/d}\) and \(X^{b/d}\). Integer coordinates in a fixed integral basis therefore give two boxes, while their projective directions satisfy explicit polynomial equations. Near a balanced point, and after choosing one of finitely many local branches, either direction determines the other analytically once \(1/n\) is specified.

We partition these directions into small paired boxes. An archimedean determinant gives a polynomial equation on the points in each pair. Components defined by an equation in only one of the two coordinate groups receive a second cut. Explicit complete intersection calculations measure the independent monomials available on the resulting sets. The distinction between a mixed equation and a pure equation is important: the latter also reduces the number of independent Taylor monomials in the local determinant.

Two further steps turn these cuts into a count. First, a reduced basis of the integer lattice associated with a direction box makes its points smaller in new integral coordinates. An exceptionally short lattice vector also limits the number of boxes in which it can occur. Second, we apply the general affine counting theorem proved in Section 3, using lower bounds for weighted spaces of polynomial functions. Its proof repeatedly uses residue classes at small auxiliary primes. Its auxiliary varieties depend on earlier choices of primes; a random deletion argument controls the primes at which their equations fail to give smooth local coordinates. This permits a uniform count through successive dimension drops.

For quartics, surfaces and curves need sharper geometric bounds. On a surface, the difficult projections have degree one or two. The quadric case supplies an additional field degree. On a plane, a deficient bound would force either a polynomial of degree at most four on a general line or a degree-two rational map on the line of directions to have four distinct critical values. On a curve, the limiting case is a polynomial of degree five with four prescribed critical values. Up to a normalization of two critical points, only finitely many such polynomials occur. Their integer values in the input interval form one set of size \(O_f(X^{1/5})\), which is removed once before summing over boxes. The remaining curve estimates and an exact finite choice of parameters give a common power saving.

Organization

Figure 1 shows how the principal estimates feed into the density theorem. Its arrows indicate uses of established modules; the order of the sections places the uniform counting statement before its arithmetic applications.

Section 2 gives the density reduction, and Section 3 proves the general counting theorem. Sections 4–6 construct the number-field boxes, determinant cuts and lattice coordinates. Section 7 establishes weighted Hilbert bounds; Sections 8 and 9 sharpen them for quartic surfaces and curves. Section 10 selects the parameters, and Section 11 proves the large-prime estimate and completes the density theorem. Section 12 gives the explicit cyclotomic density, retains a quantitative central-prime estimate, and explains the special factorization, level equation and Pell alternative.

Smooth patches with controlled equations

A patch will mean a triple \((A,G,\Delta)\), where \(A\) is an irreducible \(\mathbb Q\)-defined variety of dimension \(h\), the list \(G\) consists of \(N-h\) integer polynomials vanishing on \(A\), and \(\Delta\) is a square Jacobian minor in \(N-h\) chosen coordinate columns. It covers the points of \(A\) at which \(\Delta\ne0\). If \(h=N\), the list is empty and \(\Delta=1\). A patch need not describe \(A\) globally as a complete intersection: its equations provide local coordinates wherever its minor is nonzero.

The last convention is essential below. A prime used to define an input variety may affect that variety, but it is not separately consulted when choosing its patch list.

A determinant cut in a simultaneous residue class

For \(h\ge1\), let \(s_h(L)\) be the sum of the first \(L\) total degrees of monomials in \(h\) variables, ordered by degree and counted with their monomial multiplicities. Counting lattice points in a simplex gives \[ s_h(L)=\frac{h}{h+1}(h!)^{1/h}L^{1+1/h}+O_h(L). \tag{13}\]

We have obtained polynomial cuts whenever a sufficiently large product of primes is good for the current patch. The remaining issue is that the patch itself will depend on earlier primes. We address this by recording the entire finite sequence of evaluation ranks.

Adaptive paths and a deletion estimate

To compare a block of primes with the same block after one prime is deleted, we must define the successive patches for both choices before knowing which primes are good. We therefore construct all \(r\) stages for every point and every block choice, even when the expected dimension drops fail. At each stage we retain the whole space of bounded-degree polynomials vanishing on the current residue class. Equal-dimensional nested spaces are equal; this will make unchanged evaluation ranks force the same subsequent patches. Choosing just one vanishing polynomial would not provide that implication.

Fix a small \(\gamma>0\) and set \(h_i=r+1-i\) for \(1\le i\le r\). Choose successive ordinary degree limits \(D_i\) for the polynomial cuts. At step \(i\), take \(D_i\) large enough for Lemma 10 in parent dimension \(h_i\), with \(\nu=\gamma/2\). This can be done inductively. Intersecting with the common zero set of a polynomial space of degree at most \(D_i\), then applying Lemma 9, gives degree bounds determined by the parent bound and \(D_i\). These bounds hold even when dimension does not decrease. Consequently they cover every path of the fixed length \(r\).

Let \(\mathcal B_1,\ldots,\mathcal B_r\) be disjoint finite blocks of primes, and write \[Q_i=\prod_{j\le i}\prod_{q\in\mathcal B_j}q.\] For each \(z\in S\), define a path through all \(r\) steps as follows. Take the first patch \((A_0,G_0,\Delta_0)\) of \(Z\) covering \(z\). Given its parent patch at step \(i\), let \[ \mathcal K_i= \left\{F\in\mathbb Q[z_1,\ldots,z_N]_{\le D_i}: F(z')=0\text{ for all }z'\in S\cap A_{i-1},\ z'\equiv z\pmod{Q_i} \right\}. \tag{14}\] Record its codimension \(\rho_i\) in the polynomial space. Take the first patch \((A_i,G_i,\Delta_i)\) covering \(z\) in \(A_{i-1}\cap V(\mathcal K_i)\). Every such patch exists because the intersection contains \(z\). In particular, all \(\Delta_i(z)\) are nonzero integers of uniformly polynomial size. An exceptional alternative or a zero-dimensional parent does not stop this construction.

Adding a prime to one block makes the vector \((\rho_1,\ldots,\rho_r)\) lexicographically nonincreasing. Until its first changed entry the parent patches agree: the new residue class is a subset of the old one, so the kernels are nested, and equal codimensions make them equal. Their intersection sets and deterministic patch lists therefore agree as well. The first changed codimension can only decrease. Moreover, equality of the complete rank vectors forces equality of all kernels and patches along the paths.

Put \(M_i=\binom{N+D_i}{N}\). Encode the rank vector by the integer \[ \mathcal R=\sum_{i=1}^r\rho_i\prod_{j>i}(M_j+1), \qquad 0\le\mathcal R<B_0:=\prod_{i=1}^r(M_i+1). \tag{15}\] It is nonincreasing when a prime is added. An unchanged encoding gives identical evaluated minors along the entire path.

Good cumulative products and the counting tree

We now choose blocks for which at least half of \(S\) has enough good primes at every stage. As the scheduled dimension falls from \(r\) to \(1\), the determinant thresholds are \(\delta_r,\ldots,\delta_1\). Earlier primes remain available at later stages, so we allocate the nondecreasing cumulative exponents \(E_r,\ldots,E_1\) instead. Recall that \(E_h=\max_{h\le s\le r}\delta_s\). Put \(E_{r+1}=0\) and \[a_i=E_{h_i}-E_{h_i+1}\ge0.\] Choose a small \(\xi>0\) such that \[ \xi(E_1+r\gamma)<\gamma/2. \tag{17}\] Take \(T_0\) large in terms of \(r,B_0,\xi\), then take \(\zeta>0\) so small that \(\zeta(T_0+1)\le\gamma\). Use disjoint prime pools of cardinality tending to infinity, all between \(X^\zeta\) and \(C_rX^\zeta\). The lower and upper Chebyshev bounds for the prime-counting function give such pools for sufficiently large fixed \(C_r\): partition the primes in that interval into \(r\) pools of comparable size. Choose the blocks independently, with block \(i\) sampled as in Lemma 11, starting its size interval at \[\ell_i=\left\lceil\frac{a_i+\gamma}{\zeta}\right\rceil.\] Every allowed size \(t_i\) then satisfies \[ a_i+\gamma\le\zeta t_i\le a_i+2\gamma. \tag{18}\] All sizes are constants as \(X\) tends to infinity.

Call \(z\in S\) regular if in each block at most a fraction \(\xi\) of its primes divide any of the full-path minors at \(z\). Lemma 11, Markov’s inequality and a union bound give \[\mathbb P(z\text{ is not regular}) \le \frac r\xi\bigl(B_0/T_0+o(1)\bigr)<\frac12\] for our sufficiently large \(T_0\) and then sufficiently large \(X\). Summing over the finite set \(S\), some fixed choice of blocks makes at least \(|S|/2\) points regular. Fix those blocks.

For a regular point, all but a fraction \(\xi\) of the primes in each of the first \(i\) blocks avoid the current parent minor \(\Delta_{i-1}(z)\). Their product is at least \[\begin{align*} X^{(1-\xi)\zeta\sum_{j\le i}t_j} &\ge X^{(1-\xi)(E_{h_i}+i\gamma)} \ge X^{\delta_{h_i}+\gamma/2}. \tag{19}\end{align*}\] The first inequality uses \(\sum_{j\le i}a_j=E_{h_i}\); the last uses (17). This cumulative product includes primes chosen before the current parent was constructed.

The full paths were needed to compare prime choices in the deletion estimate. We now arrange the full paths of regular points in a tree, using the same kernels and fixed patch lists, and stop whenever a bound is already available. Start from their bounded list of initial patches; a node records its parent patch and the previous residue classes. Stop at a parent satisfying (10), charging all its points by that bound; also stop at a zero-dimensional parent. Otherwise continue as follows. We show inductively that a parent at step \(i\) has dimension at most \(h_i\).

Fix such a parent patch \((A,G,\Delta)\) of dimension \(h\le h_i\). At an auxiliary prime \(q\), its reductions with \(\Delta\ne0\) number \(O(q^h)\). Indeed, fix the \(h\) free coordinates determined by the minor. The other \(N-h\) equations have invertible Jacobian at every point being counted, so these points are isolated zeros of that square system. Bézout bounds their number by a fixed constant, even if other components of the fiber have positive dimension. Summing over the free coordinates proves the bound. It uses no assumption that the whole variety has good reduction.

For regular points, the primes of the new block at which this parent minor vanishes form a subset of size at most \(\xi t_i\). Allow all such subsets; their number is at most \(2^{t_i}\), a constant. At those primes use the trivial \(q^N\) bound and at the others use \(O(q^{h_i})\). Since \(q\le C_rX^\zeta\), the number of new simultaneous residue classes is at most \[ O\!\left(X^{\zeta t_i(h_i+N\xi)}\right) \le O\!\left(X^{(a_i+2\gamma)(h_i+N\xi)}\right). \tag{20}\] The constants from Bézout raised to \(t_i\) are still independent of \(X\). Imposing earlier congruences can only decrease this count.

Consider one resulting class containing a regular point \(z\). If the parent has dimension exactly \(h_i\), then (19) and Lemma 10 give a polynomial in \(\mathcal K_i\) that does not vanish identically on the parent. This polynomial vanishes on the entire class in \(S\cap A\): every class member has the same residues as \(z\) at the good primes and satisfies \(G=0\). Thus it is a cut for the kernel in (14), not merely for its regular points. Consequently \(A\cap V(\mathcal K_i)\) has dimension at most \(h_i-1\), and its fixed patch list has boundedly many members. If the parent had smaller dimension already, its children automatically satisfy the same bound. This proves the dimension induction.

After \(r\) stages every nonexceptional leaf is zero-dimensional and contains boundedly many points. The total branching exponent from (20) is at most \[ \sum_{i=1}^r h_i a_i +2\gamma\sum_{i=1}^r h_i +N\xi(E_1+2r\gamma). \tag{21}\] The first term telescopes to \[\sum_{h=1}^r h(E_h-E_{h+1})=\sum_{h=1}^r E_h.\] Every prefix of the tree has no larger exponent, because all the block increments and errors are nonnegative. Hence stopping at exceptional parents multiplies this bound only by \(O(X^\alpha)\), using (10); the bounded number of possible stopping depths contributes a constant.

To complete the choices for a prescribed \(\varepsilon\), first take \(\gamma\) so small that the second term of (21) is less than \(\varepsilon/4\). Choose the degree limits and hence \(B_0\). Then take \(\xi\) small enough for (17) and for the last term of (21) to be less than \(\varepsilon/4\). Take \(\alpha<\varepsilon/4\), then choose \(T_0\), \(\zeta\) and finally \(X\) as above. This order makes every degree, height and probability constant fixed before \(X\) tends to infinity. We have bounded at least half of \(S\) by the right side of (12); multiplying by two proves Theorem 8.

Uniform analytic coordinates

Split into the finitely many choices of a largest absolute coordinate of \(x\) and of \(y\), breaking ties in a fixed order. Within each chart reindex them as \(x_0,y_0\). Invertibility of the embedding matrices and (23) give \[|x_0|\asymp_f P^{1/d},\qquad |y_0|\asymp_f X/P^{k/d}.\] Put \[s_j=x_j/x_0,\qquad t_j=y_j/y_0\quad(1\le j<d), \qquad z=1/n.\] Thus \(s,t\in[-1,1]^{d-1}\). Every normalized conjugate form \(a_i(1,s)\) and \(b_i(1,t)\) has modulus bounded above and below by positive constants. The chart vectors \(s,t\) here are distinct from the scalar parameter used later to choose bidegrees.

We have reduced the selected inputs to finitely many arithmetic cases and paired projective boxes. The equations (25) retain the integer input \(n\), while the analytic descriptions provide the small variables for the determinant cuts in the next section.

The bicone and its Hilbert polynomial

Fix the distinct roots \(\theta_1,\ldots,\theta_d\), and put \(k=d-2\). In \(\mathbb C^{2d}\), with coordinate blocks \(a=(a_i)\), \(b=(b_i)\), let \(\mathcal B\) be the scheme defined by the condition \[(a_i^kb_i)_{i=1}^d \in \operatorname{span}_{\mathbb C}\{(1)_i,(\theta_i)_i\}.\] Equivalently, impose \(d-2\) independent linear relations on these products. Each relation has bidegree \((k,1)\). Write the products as \[a_i^kb_i=N_0-\theta_iL_0;\] on \(\mathcal B\), the functions \(N_0,L_0\) are fixed linear combinations of the products. The corresponding subvariety of \(\mathbb P^{d-1}\times\mathbb P^{d-1}\) will be denoted by \(\mathcal V\).

Define the one-variable polynomials \[H(t)=d!R(1,t),\qquad H_a(t)=(d-1)!(\partial_A R)(1,t),\qquad H_b(t)=(d-1)!(\partial_B R)(1,t).\] They measure the spaces of determinant columns on \(\mathcal V\) and on a pure hypersurface section, respectively.

A uniform Taylor determinant estimate

We first describe the analytic estimate used in both cuts. Let \(\ell\) column functions have uniformly bounded derivatives on the chart neighborhoods, and let the small variables be \(u_1,\ldots,u_{d-1},z\), of weights \(w,\ldots,w,1\). For a fixed finite set of evaluation points, suppose that the rank of the evaluation columns of monomials of weight at most \(T\) is at most \[aT^r+O(T^{r-1}+1),\qquad a>0.\] If \(\ell\) such monomial columns are independent, and their weights in increasing order are \(\lambda_j\), then \(j\leq a\lambda_j^r+O(\lambda_j^{r-1}+1)\). Inverting and summing gives \[ \sum_{j=1}^{\ell}\lambda_j \geq \frac{r}{r+1}a^{-1/r}\ell^{1+1/r}-O(\ell). \tag{29}\] The constants are uniform whenever the rank estimate is uniform.

Expand each analytic column at the box base in the original monomials in \(u,z\), to a fixed total order \(K\). A polynomial summand in the multilinear expansion of the determinant vanishes if its monomial evaluation columns are dependent. Otherwise (29) bounds its total weight, and hence its absolute value by a constant times \(X\) to the negative of that weight. The Taylor coefficients are uniformly bounded. The remainder in a column is \(O(X^{-\min(w,1)(K+1)})\). After fixing the column degrees and \(\ell\), choose \(K\) so large that every determinant term containing a remainder has a stronger bound than the required one. There are then only boundedly many terms. Thus the normalized determinant is \[ \ll X^{-S},\qquad S=\frac{r}{r+1}a^{-1/r}\ell^{1+1/r}-O(\ell). \tag{30}\] This argument requires only exact dependence of evaluation columns, not any bound on coefficients expressing that dependence.

The first cut and its components

Choose large integer bidegrees with \(B/A=t_*\), and choose a basis of the degree-\((A,B)\) coordinate space of \(\mathcal V\) from raw monomials in the integral coordinates \(x,y\). By Lemma 14, its size is \[\ell=\frac{H(t_*)}{d!}A^d+O(A^{d-1}).\] For any \(\ell\) counted points, the raw evaluation determinant is an integer. Divide each row by \(x_0^Ay_0^B\), using the nonzero chart coordinates. If the determinant is nonzero, its normalized absolute value is at least \[ c_{A,B}\,X^{-\ell A(\eta_++b_+t_*)/d}, \qquad c_{A,B}>0. \tag{31}\] The constant can be chosen uniformly in the box.

Normalized monomials are analytic functions of the chart variables and \(z\). Without an extra relation among the small variables, their monomial count is \[\frac{T^d}{d!w^{d-1}}+O(T^{d-1}+1).\] Equations (29)–(30) therefore give \[S=\frac d{d+1}(d!w^{d-1})^{1/d}\ell^{1+1/d}-O(\ell), \qquad \frac S\ell =\frac d{d+1}\bigl(w^{d-1}H(t_*)\bigr)^{1/d}A+O(1).\] The first inequality in (28) makes this exponent larger than the one in (31), first for sufficiently large fixed \(A,B\), then for sufficiently large \(X\). Every such determinant vanishes. The evaluation map therefore has rank less than \(\ell\), also when there are fewer than \(\ell\) points. Its kernel gives a bihomogeneous equation vanishing on all the counted points but not identically on \(\mathcal V\).

We must describe the actual components of this cut before using their Hilbert functions. On the open where all \(a_i\ne0\), substitute \(b_i=(N_0-\theta_iL_0)/a_i^k\), clear denominators, and factor. Each relevant irreducible factor \[F(a,N_0,L_0)=\sum_I a^If_I(N_0,L_0)\] is bihomogeneous, say of degrees \((l,s_0)\) in \(a\) and \((N_0,L_0)\). Indeed, the two scaling actions preserve each irreducible factor up to scalar. Factors \(a_i\) or \(L_0\) do not meet the open of counted points and can be omitted. Returning to the \((a,b)\)-coordinates replaces \(N_0-\theta_iL_0\) by \(a_i^kb_i\), so the substituted terms may acquire common powers of \(a_i\). We extract these powers and then verify that the remaining equation has no extra boundary component. For each nonzero \(f_I\), set \[\nu_{i,I}=\operatorname{ord}_{N_0-\theta_iL_0}f_I,\qquad e_i=\min_I(I_i+k\nu_{i,I}).\] Writing \(f_I=\prod_i(N_0-\theta_iL_0)^{\nu_{i,I}}g_I\), the expression \[ C(a,b)=\sum_I a^{I+k\nu_I-e}b^{\nu_I}g_I(N_0,L_0) \tag{32}\] is a polynomial: every exponent of \(a\) is nonnegative. It represents \(F/\prod_i a_i^{e_i}\) on the bicone, and has bidegree \[(\lambda,s_0),\qquad \lambda=l+ks_0-\sum_i e_i.\]

This extension has no unwanted boundary component. On \(a_i=0\) with nonzero product vector, set \(L_0=1\). Substitution for the other \(b_j\) leaves, precisely when \(I_i+k\nu_{i,I}=e_i\), a nonzero constant times \[a_{-i}^{I_{-i}-e_{-i}}b_i^{\nu_{i,I}}.\] These Laurent monomials are distinct: their exponents determine \(I_{-i}\) and \(\nu_{i,I}\), and then \(I_i=e_i-k\nu_{i,I}\). Thus the restriction of \(C\) is nonzero. It reduces this boundary stratum to dimension at most \(d\); the zero-product boundary already has dimension \(d\). The open defined by \(F\) is integral and generically reduced of dimension \(d+1\). Since \(C\) is a nonzero element of the prime bicone ring, adding it preserves the complete-intersection property. The boundary calculation and the criterion in the proof of Lemma 14 now prove that the new ideal is prime.

The Hilbert series acquires the factor \(1-z_1^\lambda z_2^{s_0}\). Its leading term in bidegree \((A,B)\) is \[ (\lambda\partial_A+s_0\partial_B)R(A,B), \tag{33}\] with error \(O(A^{d-2})\) at fixed \(B/A\). The nonnegative integers \(\lambda,s_0\) are not both zero. If both are positive we retain this mixed component. All degrees remain bounded, by the degree of the first determinant equation and the fixed substitutions.

The second cut on pure components

Suppose instead that \(C\) uses only one block, with degree \(c_0>0\). Use the \(s\)-chart for a pure \(a\) equation and the \(t\)-chart for a pure \(b\) equation. In the latter case, partition the counted points according to the boundedly many reverse branches at the common \(t\)-box base supplied by Lemma 13, and apply the following determinant argument to each subset. Thus all rows of each determinant use the same analytic chart map. This only multiplies the number of resulting sets by a bounded factor. Dehomogenization and translation to the box base give an exact polynomial relation \[Q_0(u_1,\ldots,u_{d-1})=0,\qquad 1\leq\deg Q_0\leq c_0.\] A nonzero constant relation gives no points. No bound on the coefficients of \(Q_0\) is imposed.

Here is the needed coefficient-uniform rank estimate. Put \(r=d-1\) and \(e=\deg Q_0\). Multiplication by \(Q_0\) injects polynomials of degree at most \(q-e\) into polynomials of degree at most \(q\), and its image evaluates to zero. Therefore the evaluation rank of the latter space is at most \[D_e(q)=\binom{q+r}{r}-\binom{q-e+r}{r},\] with the second term zero when \(q<e\). At each power \(z^j\), apply this bound with \(q=\lfloor(T-j)/w\rfloor\). Since \(D_e(q)=e q^{r-1}/(r-1)!+O_{r,c_0}((q+1)^{r-2})\), summing gives \[N(T)\leq \frac{c_0}{r!w^{r-1}}T^r+O_{r,c_0,w}(T^{r-1}+1).\] This bound involves no coefficients of the relation. Inserting it into the Taylor argument above gives \[S\geq\frac r{r+1} \left(\frac{r!w^{r-1}}{c_0}\right)^{1/r} \ell^{1+1/r}-O(\ell).\] In particular, we have never solved the moving equation for one variable or reduced Taylor monomials by division by a coefficient.

Keep the first-cut degrees fixed, so that \(c_0\) ranges over a bounded set of positive integers. Choose new, sufficiently large bidegrees \(A,B\), again with \(B/A=t_*\), for the second determinant. By (33), a basis of raw monomials on a pure \(a\) component has size \[\ell=\frac{c_0H_a(t_*)}{r!}A^r+O(A^{r-1}).\] Consequently \[\frac S\ell\geq \frac r d\bigl(w^{r-1}H_a(t_*)\bigr)^{1/r}A-O(1).\] The factor \(c_0\) cancels. The identical calculation for a pure \(b\) component uses \(H_b\). The second inequality in (28) contradicts (31) in both cases. The resulting equation is nonzero on the component, so it cuts its biprojective dimension from \(d-1\) to at most \(d-2\). The Hilbert series and errors depend only on bounded bidegrees, and the analytic bounds come from the ambient chart maps. Thus the second-cut degrees can be fixed uniformly for every pure component.

Finally, on the counted affine locus \(L_0=1\), the map to \(\mathcal V\) has one-dimensional scaling fibers: \(a\mapsto ca\) requires \(b\mapsto c^{-k}b\). Mixed components therefore have affine dimension \(d\), and second-cut components have dimension at most \(d-1\). In the mixed case the equation on the nonzero-\(a\) open is \(F(a,n,1)=0\). It is irreducible, since a factorization would rehomogenize to one of \(F\), and its degree in \(n\) is exactly \(s_0\), since \(L_0\) does not divide \(F\). Since \(s_0\geq1\), its coefficient polynomials in \(a\) have no common nonconstant factor; such a factor would contradict irreducibility. Their common zero set therefore has codimension at least two in the \(a\)-space whenever it is nonempty. Taking closures preserves these dimensions and bounded degrees. This proves Proposition 15.

A basis bound with uniform constants

Grouping the paired boxes

Fix a maximal-coordinate chart from Section 4, and reindex its coordinates as \(x=(x_0,\ldots,x_{d-1})\), \(y=(y_0,\ldots,y_{d-1})\). Thus \(s_i=x_i/x_0\) and \(t_i=y_i/y_0\) for \(1\le i<d\). Let \(\eta_+,b_+>0\) be fixed upper exponents for which \[|x_i|\ll X^{\eta_+/d},\qquad |y_i|\ll X^{b_+/d}.\] Choose fixed \(w,\tau>0\) with \[ m=(d-1)w<\min(\eta_+,b_+),\qquad u=\frac{\eta_+-m}{d},\qquad v=\frac{b_+-m}{d}. \tag{34}\] Use a prime \(M\asymp X^w\) for the approximation grid. Such a prime exists by Bertrand’s postulate. All constants in this section are uniform in the grid boxes and in \(X\); they may depend on the fixed parameters, the dimension, and the polynomial.

Linear changes preserve dimensions and degrees of all the cut sets. They also preserve the weighted polynomial spaces within each coordinate block: every \(a_i\) is a linear combination of the \(x'_i\), which all have weight \(U\), and every \(b_i\) is a linear combination of the \(y'_i\), which all have weight \(V\). This assertion concerns the polynomial filtration, irrespective of the coefficients of the linear forms. The numerical size bounds used for integral counting are those on \(x',y'\) in (36).

Removing infinite fibers in the quartic case

For \(d=4\), a later Hilbert estimate requires \(n\) to be algebraic over the functions \(a_1,\ldots,a_4\). We arrange this by removing a small set of possible \(x'\) from each initial cut set of Proposition 15.

Proposition 17 supplies the integral boxes for the affine counting theorem, and Lemma 18 supplies the additional projection property needed in the quartic case. The next sections establish the Hilbert bounds on the sets inside these boxes; the final summation will combine their cost with the factor \(X^{m-dj}\) in (35).

Extracting a polynomial equation

Factor \[f_I(n)=\prod_{i=1}^h(n-\theta_i)^{\nu_{i,I}}g_I(n).\] As in (32), define \[ C_1(a,b)= \sum_I a^{I+k\nu_I-e}b^{\nu_I} g_I(a_1^kb_1+\theta_1). \tag{40}\] All exponents of \(a\) are nonnegative. Impose, in addition, the \(h-1\) equations \[ a_i^kb_i+\theta_i=a_1^kb_1+\theta_1 \qquad(2\leq i\leq h). \tag{41}\] On the open where all \(a_i\ne0\), these equations and \(C_1=0\) have coordinate ring \[\mathbb C[a_1^{\pm1},\ldots,a_h^{\pm1},n]/(F_1),\] which is integral of dimension \(h\).

The boundary has smaller dimension. If \(a_i=0\), then \(n=\theta_i\), every other \(a_j\ne0\), and \(b_j=(\theta_i-\theta_j)/a_j^k\). Before imposing \(C_1\), the free coordinates are \(b_i\) and the \(h-1\) nonzero \(a_j\), giving dimension \(h\). The terms of \(C_1\) surviving there have \(I_i+k\nu_{i,I}=e_i\) and are nonzero scalar multiples of \[a_{-i}^{I_{-i}-e_{-i}}b_i^{\nu_{i,I}}.\] Their exponents determine \(I\), so these Laurent monomials are distinct. Thus \(C_1\) is nonzero on this stratum and reduces its dimension to at most \(h-1\). This checks every affine boundary stratum, since two different \(a_i\) cannot vanish when the \(\theta_i\) are distinct.

Every term in (40) has weight at most the number \(D\) in (38). The bound is attained without cancellation. To see this, retain the highest weighted parts in (41); they say that the products \(a_i^kb_i\) all equal a common variable \(q\). On their nonzero-\(a\) open, the weight-\(D\) part of \(C_1\) restricts to \[ a^{-e}\operatorname{in}_{(U,kU+V)}F_1(a,q). \tag{42}\] Here \(\operatorname{in}_{(U,kU+V)}\) means the sum of terms of largest weight when each \(a_i\) has weight \(U\) and \(q\) has weight \(kU+V\). The expression is a nonzero Laurent polynomial: distinct monomials \(a^Iq^j\) remain distinct after division by \(a^e\). This also follows directly from the extracted-factor formula, since constants \(\theta_i\) disappear in the highest-weight terms. Consequently \(C_1\) has exact weight \(D\). It is nonconstant, since otherwise the nonempty open hypersurface defined by \(F_1\) would be empty. Hence \[ D\geq\min(U,V)>0. \tag{43}\]

The complete intersection at weighted infinity

Choose a positive integer \(L_{\mathrm{wt}}\) such that \(A=L_{\mathrm{wt}}U\) and \(B=L_{\mathrm{wt}}V\) are integers, and set \[Q=kA+B,\qquad E=L_{\mathrm{wt}}D.\] Homogenize (41) and \(C_1\) with a variable \(t\) of weight one, giving \(h\) homogeneous equations in \(2h+1\) variables. On \(t\ne0\), scaling to \(t=1\) identifies the scheme with the affine one times \(\mathbb C^\times\). Its nonzero-\(a\) open is integral and generically reduced of dimension \(h+1\). Its remaining affine strata have dimension at most \(h\), by the preceding boundary calculation.

At \(t=0\), the products \(a_i^kb_i\) must coincide. If all \(a_i\ne0\), the variables \(a_1,\ldots,a_h,q\) are free before the last equation, and (42) is nonzero. This part of infinity has dimension at most \(h\). If some \(a_i=0\), the common product is zero, so \(a_j^kb_j=0\) for every \(j\). This locus is a union of coordinate \(h\)-planes. Its dimension is \(h\), even if the last equation vanishes on an entire such plane.

It follows that the homogeneous system has dimension \(h+1\), and no top-dimensional component is contained in either boundary. The complete-intersection criterion used in Lemma 14 therefore makes its ideal prime. In particular, it is saturated with respect to \(t\). Dehomogenization gives precisely the prime ideal of \(Y_{\mathrm{pair}}\): on the nonzero-\(a\) open the descriptions agree, and the boundary has smaller dimension.

The degree-\(N\) part of this graded quotient is the space of affine functions represented by monomials of weight at most \(N\) in the scaled weights \(A,B\). Surjectivity follows by homogenizing each such monomial with a power of \(t\). For injectivity, a homogeneous element vanishing after dehomogenization vanishes after localization at \(t\), and saturation implies that it was zero in the graded quotient. Thus the Hilbert series of these filtered dimensions is \[ \frac{(1-z^Q)^{h-1}(1-z^E)} {(1-z)(1-z^A)^h(1-z^B)^h}. \tag{44}\]

At \(z=1\) this series has a pole of order \(h+1\), with leading factor \(Q^{h-1}E/(A^hB^h)\). We also check all other roots of unity. If only one of \(1-z^A,1-z^B\) vanishes, its denominator order is \(h\). If both vanish at a nontrivial root \(\zeta\), its order divides \(\gcd(A,B)\), and therefore divides \(Q\) and \(E\): the latter is a monomial weight of the nonzero polynomial \(C_1\). All \(h\) numerator factors then vanish, reducing the denominator order \(2h\) to at most \(h\). The extra factor \(1-z\) is nonzero at \(\zeta\). Partial fractions consequently give \[\frac{Q^{h-1}E}{A^hB^h}\frac{N^h}{h!} +O(N^{h-1}+1)\] for the coefficient of \(z^N\). Taking \(N=\lfloor L_{\mathrm{wt}}T\rfloor\) and rescaling proves (39). In particular, lower-dimensional coordinate planes at infinity do not change its leading coefficient or introduce a periodic term of order \(T^h\).

Uniformity and function-field extensions

The degree of \(F_1\) is bounded in terms of \(d,k\) and the degree of \(Y\). Indeed, \(n=a_1^kb_1+\theta_1\) is a polynomial of fixed degree. Form the graph of the map to the selected \(a_i\) and \(n\), then project linearly; the usual degree bounds for graphs and projections give the assertion. They do not involve coefficient heights. For fixed \(U,V\) and a bound on that degree, only finitely many numerator degrees \(E\) can occur in (44). Its coefficient errors are therefore uniform. This proves the required uniformity even for moving equations and moving blockwise linear changes of coordinates.

Pullback from the pair image is injective and preserves weights, so its Hilbert function is a lower bound for that of \(Y\). Let \(K_{\mathrm{pair}}=\mathbb C(Y_{\mathrm{pair}})\). If \([\mathbb C(Y):K_{\mathrm{pair}}]\geq2\), one of the original \(a\) or \(b\) coordinates, say \(\xi\), lies outside \(K_{\mathrm{pair}}\), since these coordinates generate \(\mathbb C(Y)\). Its weight \(\omega\) is \(U\) or \(V\). The two spaces consisting of pair-image functions of weight at most \(T\), and \(\xi\) times such functions of weight at most \(T-\omega\), are linearly independent on \(Y\): a relation \(A+\xi B=0\) with \(A,B\in K_{\mathrm{pair}}\) would contradict \(\xi\notin K_{\mathrm{pair}}\) unless \(A=B=0\). Hence \[ H_Y(T)\geq H_{Y_{\mathrm{pair}}}(T)+H_{Y_{\mathrm{pair}}}(T-\omega) \tag{45}\] for \(T\geq\omega\). The bounded shift leaves the leading coefficient unchanged and only changes the uniform \(O(T^{h-1}+1)\) term. This proves the asserted doubling and completes Proposition 19.

The general dimensional thresholds

We record the consequences needed before the special quartic surface argument. Here \(Y\) is an irreducible subvariety of one of the initial affine sets obtained in Proposition 15. If \(Y\) has dimension \(d\), it must be one of the mixed affine components of Proposition 15; the second-cut sets have smaller dimension. In this case the equation is \(F(a,n,1)\), with \(F\) of bidegree \((l,s_0)\) and saturated bidegree \((\lambda,s_0)\), where \(\lambda,s_0\geq1\). Homogeneity in \(a\), and the fact that its degree in \(n\) is exactly \(s_0\), give \[D=lU+(kU+V)s_0-U\sum_i e_i =\lambda U+s_0V\geq U+V.\] Thus (39) gives a lower bound with leading coefficient \[\frac{(kU+V)^{d-1}(U+V)}{U^dV^d}.\] In every dimension \(2\leq h\leq d-1\), the general bound (43) gives the coefficient \[\frac{(kU+V)^{h-1}\min(U,V)}{U^hV^h}.\]

If instead we can select \(h\) algebraically independent \(a\)-coordinates, the same pair-image construction applies. Its polynomial \(C_1\) must involve a \(b\)-coordinate; otherwise it is a nonzero relation among those independent \(a\)’s. Consequently \(D\geq V\), giving coefficient \[\frac{(kU+V)^{h-1}V}{U^hV^h}.\] After the quartic infinite-fiber removal in Lemma 18, \(n\) is algebraic over the \(a\)-coordinate field on every remaining subvariety. That field has transcendence degree \(h\), so the independent selection exists. In particular, this last coefficient supplies the quartic \(h=3\) bound. The next section sharpens the remaining quartic surface case.

The surface bound

One global set of quintic values

The curve argument will encounter degree-five polynomials with four prescribed critical values. Their coefficients can initially depend on the box and on the curve. The following normalization places all of them in one fixed finite family before any counting is done.

Coverage and the remaining exponent margins

The transition function \(\phi_d(\eta)=\eta(1-k\eta/d)\) is nonincreasing for \(\eta\ge1\), since \[\phi_d'(\eta)=1-\frac{2k\eta}{d} \le\frac{4-d}{d}\le0.\] Consequently all intervals before the stopping index are covered by Proposition 24, and every subsequent \(\eta\le d/k\) satisfies the larger-\(\eta\) condition \(\phi_d(\eta)<2495/10000\). The last closed interval ends at \(z_d/K_0\), so the two ranges meet without a gap. In the quartic case, the last \(\mathsf Q\) interval ends at \(4/3\) and the next interval uses the ordinary thresholds; either interval covers their common endpoint.

The lattice ranges of Proposition 17 have \(U=u+j+\tau\), \(V=v+j+\tau\) and box factor \(X^{m-dj}\). By (62), their combined exponent is at most \[ m-dj+\sum_{h=1}^dE_h(U,V) \le m+\sum_{h=1}^dE_h(u,v)+d\tau <\frac{24979}{25000}<1. \tag{65}\] The estimate uses only the advertised \(999/1000\) bound and \(d\le8\); Table 1 gives stronger bounds. For mode \(\mathsf Q\), the removed base fibers have combined exponent \[ m-4j+2U=m+2u-2j+2\tau <\frac{3122}{3125}<1. \tag{66}\] In particular, both inequalities leave positive room for the arbitrarily small loss in the affine counting theorem.

The direct larger-\(\eta\) argument uses a fixed finite set of positive rational coordinate weights even at the terminal point \(b=d-k\eta=0\). The available product gap is \[ \left(\frac{4999}{10000}\right)^2-\frac{2495}{10000} =\frac{40001}{100000000}>0. \tag{67}\] An explicit choice on \(1\le\eta\le d/k\) is, with \(e=1/10000\), \[b'=e(\lfloor b/e\rfloor+1),\qquad \eta'=e(\lfloor\eta/e\rfloor+1).\] These are strict positive upper weights, drawn from a finite set. Since \(b+\eta\le3\) and \(d\ge4\), whenever \(b\eta/d\le2495/10000\) they satisfy \[\frac{b'\eta'}d \le\frac{2495}{10000}+\frac{3e+e^2}{4} <\left(\frac{4999}{10000}\right)^2.\] Also include the pair \((b',\eta')=(e,d/k+e)\), which covers \(d/k\le\eta<d/k+e\). This includes every fixed-coefficient \(O_f(1/\log X)\) overshoot once \(X\) is sufficiently large. Thus the surface threshold in the direct argument is strictly below \(4999/10000\), including the endpoint range. Together with the curve threshold \(1/2\), the exponent from the two cumulative maxima is strictly below \(9999/10000\). These fixed margins are the numerical input for the final tail summation.

The range handled by the original surface

Suppose first that \[ \eta\left(1-\frac{k\eta}{d}\right)\le\frac{2495}{10000}. \tag{68}\] The chosen triples \[(x,y,z)=\left(n,\frac{f(n)}{p^k},p\right)\] are integer points on the irreducible affine surface \[ f(x)=yz^k. \tag{69}\] Its irreducibility follows from Gauss’s lemma, since it is primitive and linear in \(y\) over \(\mathbb C[x,z]\): the coefficients \(z^k\) and \(f(x)\) are coprime. The coordinate bounds are \(O_f(X)\), \(O_f(X^b)\) and \(O_f(X^\eta)\).

Use the finite positive rational upper weights \(b',\eta'\) constructed at the end of Section 10. They satisfy \[ \sqrt{b'\eta'/d}<0.4999. \tag{70}\] The construction covers \(b=0\) and the possible \(O_f(1/\log X)\) excess beyond \(d/k\), for all sufficiently large \(X\). Work separately with these finitely many choices.

For weights \((1,b',\eta')\), the space of polynomial functions on (69) has filtered dimension \[ H(T)=\frac{D}{b'\eta'}\frac{T^2}{2}+O(T),\qquad D=\max\{d,b'+k\eta'\}. \tag{71}\] Indeed multiplication by the defining polynomial increases weight by exactly \(D\), since its weighted highest part is nonzero and the polynomial ring is a domain. Choose a positive integer \(L\) so that \(A=L\), \(B=Lb'\), \(C=L\eta'\) and \(E=LD\) are integers. Weighted homogenization, with a variable of weight one, gives the filtered Hilbert series \[\frac{1-t^E}{(1-t)(1-t^A)(1-t^B)(1-t^C)}.\] The pole at one has order three and leading coefficient \(E/(ABC)\). Every other root of unity has pole order at most two: if its order divides all three coordinate weights \(A,B,C\), it also divides the monomial weight \(E\), so the numerator cancels a denominator factor. Extracting the coefficient of \(t^{\lfloor LT\rfloor}\) and rescaling gives (71), with the stated \(O(T)\) error. Consequently the surface threshold in Theorem 8 may be taken to be \[\delta_2=\sqrt{b'\eta'/D}<0.4999.\]

Consider an irreducible curve defined over \(\mathbb Q\) on this surface and meeting the selected points. A constant \(x\) contributes at most one selected triple. Otherwise \(\mathbb C(x)\) is a subfield of its function field. If the extension has degree at least two, one of the other coordinate functions lies outside \(\mathbb C(x)\). Multiplying powers of \(x\) by that function and by one gives two independent families, so \(H(T)\ge2T-O(1)\).

In the remaining case \(y\) and \(z\) are rational functions of \(x\). They are defined over \(\mathbb Q\): the curve is defined over \(\mathbb Q\) and the functions are uniquely determined by \(x\). Their degrees are bounded in terms of the curve degree, by projection. A nonpolynomial function is covered by Lemma 22 and contributes \(O_\varepsilon(X^\varepsilon)\) points. If both are polynomials, the identity \(f(x)=y(x)z(x)^k\) and the absence of repeated roots of \(f\) force \(z\) to be constant. At selected points this constant is a prime \(p>X\). Outside the fixed bad primes, \(f\) has at most \(d\) roots modulo \(p^k\), and \(p^k>X\); hence this curve contributes \(O_f(1)\) inputs in \((X,2X]\). These alternatives are uniform for bounded curve degrees.

An irreducible two-dimensional subvariety of (69) is the surface itself. Its Hilbert bound and the curve alternatives above, including the bounded constant-\(x\) and constant-\(z\) cases, therefore verify the hypotheses of Theorem 8 with \(\delta_1=1/2\). Since \(\delta_2<\delta_1\), that theorem bounds the number of triples by \[O_{f,\varepsilon}\bigl(X^{\delta_2+1/2+\varepsilon}\bigr).\] The finite list of choices in (70) permits one sufficiently small \(\varepsilon>0\) and one positive power saving.

The number-field range

For all other \(\eta\), use one of the finite parameter intervals from Proposition 24. Denote its parameters by \(t_*,w,m,u,v\) and use the thresholds \(\delta_h(U,V)\) defined there. In particular, \[m=(d-1)w,\qquad 0<m<\min(\eta_-,b_-),\qquad u=\frac{\eta_+-m}{d},\quad v=\frac{b_+-m}{d},\quad \tau=\frac1{50000}.\] The finite interval list covers the complement of (68); its stopping endpoints lie strictly inside the latter range. This supplies coverage at all boundaries.

For \(d=4\), remove once the set of exceptional values in Proposition 21. Its total size is \(O_f(X^{1/5})\), independently of the subsequent prime intervals, boxes or curves. Lemma 12 and Lemma 13 place the remaining chosen inputs in paired arithmetic boxes. The strict inequalities in the parameter certificate permit the cuts of Proposition 15.

For each box pair, the resulting affine sets have dimension at most \(d-1\), except for the mixed-cut components of dimension \(d\). All their degrees and their number per pair are bounded before \(X\) varies. Group the pairs as in Proposition 17. There are only boundedly many indices \(j\ge0\), and a group has \[ O_f(X^{m-dj})\text{ pairs},\qquad |x'_i|\ll_f X^U,\quad |y'_i|\ll_f X^V, \quad U=u+j+\tau,\quad V=v+j+\tau. \tag{72}\] The coordinate changes are integral and unimodular. Their coefficients may vary; the preceding Hilbert statements are uniform under these changes and fixed degree bounds.

In quartic mode \(\mathsf Q\), first delete the infinite-fiber points of Lemma 18 from each initial set. The cost per box is \(O_f(X^{2U})\). On the remaining point set, Proposition 19, Proposition 20 and Proposition 23 give, for every irreducible \(\mathbb Q\)-defined subvariety that can occur in the affine counting theorem, either its uniform small-count alternative or \[H_A(T)\ge\frac{T^h}{h!\,\delta_h(U,V)^h}-O(T^{h-1}), \qquad h=\dim A>0.\] Outside mode \(\mathsf Q\) the general Hilbert and curve statements give the same assertion with the corresponding thresholds. A constant input or constant first coordinate tuple has the already proved bounded count. The deletion of the quintic values is precisely what supplies the improved quartic curve threshold.

Set \(E_h(U,V)=\max_{h\le s\le d}\delta_s(U,V)\). Theorem 8, with ambient dimension \(2d\) and dimension bound \(d\), bounds each remaining set by \(O_{f,\varepsilon}(X^{\sum_hE_h(U,V)+\varepsilon})\). Its hypotheses have been checked for every bounded degree, so they apply to the moving subvarieties introduced by that theorem, not just to the initial cut components. Multiplying by (72), and applying the shift inequality in the parameter certificate, gives the exponent \[\begin{align*} m-dj+\sum_{h=1}^d E_h(u+j+\tau,v+j+\tau)+\varepsilon &\le m-dj+\sum_{h=1}^d\bigl(E_h(u,v)+j+\tau\bigr)+\varepsilon\\ &=m+\sum_{h=1}^dE_h(u,v)+d\tau+\varepsilon. \end{align*}\] By (65), this is less than \(24979/25000+\varepsilon\). A sufficiently small fixed \(\varepsilon\) therefore leaves a uniform positive gap below one.

The quartic infinite-fiber cost, including the number of boxes, has exponent \[m-dj+2(u+j+\tau)<3122/3125<1\] by (66). Thus it also has a uniform positive saving. The finite number of parameter intervals, boundedly many lattice groups and bounded component multiplicities affect only the implied constant.

Completion

The determinant degrees, patch degrees, Hilbert error constants and counting tolerances are fixed using the strict margins before \(X\) tends to infinity. There are finitely many parameter intervals; take the minimum of their savings and that of the direct surface range. Increasing one threshold for \(X\) absorbs all bounded constants. Summing over the \(O_f(\log X)\) dyadic prime intervals then preserves a smaller positive power saving. The single quartic exceptional set is also smaller than this bound after decreasing the saving if necessary. This proves Proposition 4.

Finally, Proposition 5 applies to a sign-normalized primitive part of each polynomial in Theorem 1. It retains the exact local factors of the original polynomial, including its content primes, and gives the asserted positive Euler-product asymptotic.

Local factors and the dyadic conclusion

The central prime range admits a stronger error estimate than the positive-proportion conclusion by itself requires. We retain it explicitly, including the small extension below the input scale.

An explicit factorization and a Pell estimate

Write \(\zeta=\exp(2\pi i/8)\) and \(R=\mathbb Z[\zeta]\). In this case the ideal-class correction in Lemma 12 can be omitted: one can take \(\mu=1\) and obtain primitive coefficient vectors. The following direct argument records why.

The ring \(R=\mathbb Z[i]+\zeta\mathbb Z[i]\) is Euclidean for the absolute field norm. Given \(a+b\zeta\in\mathbb Q(\zeta)\), round \(a,b\in\mathbb Q(i)\) to Gaussian integers, with errors \(c,d\) satisfying \(|c|^2,|d|^2\le1/2\). Then \[|N_{\mathbb Q(\zeta)/\mathbb Q}(c+d\zeta)|=|c^2-id^2|^2<1.\] The triangle inequality gives strictness unless both errors have squared modulus \(1/2\). In that remaining case their real and imaginary parts have absolute value \(1/2\), so \(c^2\) and \(id^2\) are perpendicular and the squared modulus is \(1/2\). Euclidean division implies that every ideal of \(R\) is principal.

If \(p^2\mid n^4+1\), evaluation \(\zeta\mapsto n\) defines surjections \(\phi_j:R\to\mathbb Z/p^j\mathbb Z\) for \(j=1,2\). Write \(\ker\phi_1=(A)\). Its index is \(p\), so \(N(A)=p\); the norm is positive because the embeddings occur in complex conjugate pairs. The inclusion \((A)^2\subseteq\ker\phi_2\) is an equality by their indices. It follows that \[ A^2B=n-\zeta,\qquad A,B\in R,\qquad N(A)=p. \tag{73}\] The unit \(1+\sqrt2\), where \(\sqrt2=\zeta-\zeta^3\), has absolute values \(1+\sqrt2\) and \((1+\sqrt2)^{-1}\) at the two complex places. Multiplication by a suitable power of this unit balances the two absolute values of \(A\). Thus, if \(N\le n\le2N\) and \(P\le p<2P\), \[|\sigma(A)|\asymp P^{1/4},\qquad |\sigma(B)|\asymp NP^{-1/2}\] at every embedding. Both coefficient vectors in the basis \(1,\zeta,\zeta^2,\zeta^3\) are primitive, since a common integer divisor of either vector would divide the coefficient \(-1\) of \(\zeta\) in \(A^2B\).

A fixed projective \(B\)-direction has at most two primitive integral representatives and hence fixes \(D=N(B)\). Taking norms in (73) then gives the same Pell equation with \(q=p\). A fixed \(A\)-direction fixes \(p=N(A)\) and one residue class for \(n\) modulo \(p^2\), so it contributes \(O(1+N/p^2)\) inputs. These are special projective-direction multiplicity bounds. The general affine argument instead selects one representation per input and uses the fixed-\(x\) bound in Lemma 12.

Directions and the level equation

For coefficient vectors \(A,B\), let \(T_0,S,C_2,C_3\) be the coefficients of \(1,\zeta,\zeta^2,\zeta^3\) in \(A^2B\). Their projective directions lie on \[\mathcal V_8=\{C_2=C_3=0\}\subseteq\mathbb P^3\times\mathbb P^3,\] while the actual integral representatives satisfy \[ T_0=n,\qquad S=-1. \tag{74}\] If \(H,J\) denote the hyperplane classes of the two factors, the complete-intersection calculation in Lemma 14 becomes \[[\mathcal V_8]=(2H+J)^2,\qquad (H^4,H^3J,H^2J^2,HJ^3,J^4)_{\mathcal V_8}=(0,1,4,4,0).\] On the split-coordinate torus, with conjugates \(\zeta_i\), its points satisfy \(A_i^2B_i=T_0+S\zeta_i\). Thus the divisor \(S=0\) is, on that open set, the inverse-square graph \([B_i]=[A_i^{-2}]\). The analytic coordinate \(-S/T_0\) is exactly \(1/n\) at the counted points. This identifies the concrete direction geometry behind the paired charts of Section 4.

There is also a determinant saving obtained directly from \(S=-1\). We give its column-selection form; it explains why that equation contains more information than its projective zero locus.

Let \(D\subseteq\mathcal V_8\) be an integral direction variety of dimension \(r\ge1\) meeting \(\{S\ne0\}\), and write \(\mathcal R_D(K,L)\) for the restrictions of bihomogeneous forms of bidegree \((K,L)\). Suppose a homogeneous polynomial \(v_0(k,l)\) of degree \(r\), with nonnegative mixed-degree coefficients, supplies the lower bound \[ \dim\mathcal R_D(Tk,Tl) \ge\frac{v_0(k,l)}{r!}T^r-O(T^{r-1}) \tag{75}\] on each fixed positive rational ray. One can use the actual mixed-degree polynomial, or coordinatewise lower bounds for its coefficients. For bounded geometric degree, the error can be taken uniformly on any fixed finite list of rays. Indeed, after the corresponding projective embedding, linear Noether normalization of the coordinate ring gives a polynomial subring whose field extension degree is the projective degree \(e\). Monomials of degree at most \(e-1\) contain a field basis: their spans increase strictly until they fill the extension. After multiplication by powers of one normalization coordinate, these give a graded injection of \(e\) copies of that polynomial ring, shifted by \(e-1\). Its degree-\(T\) dimension is \(e\binom{T-e+1+r}{r}\), which proves the required uniform lower bound.

Suppose the integral representatives in polynomial-height lattice bases satisfy \(|A|\ll N^h\), \(|B|\ll N^g\), where \(h,g\ge0\). For positive rational \(k,l\), put \(a=\min(k/2,l)\) and assume \(v_0(k,l)>0\). A nested selection of integral columns has average logarithmic size, divided by the common bidegree scale \(T\), at most \[ kh+lg-(2h+g)\int_0^a \frac{v_0(k-2j,l-j)}{v_0(k,l)}\,dj+\varepsilon. \tag{76}\] Here \(\varepsilon>0\) is arbitrarily prescribed: first fix a sufficiently fine mesh and then a sufficiently large \(T\), before \(N\) grows.

To prove this, fix a rational mesh \(0=j_0<\cdots<j_s<a\) and consider the nested subspaces \[S^{j_iT}\mathcal R_D(T(k-2j_i),T(l-j_i)) \subseteq\mathcal R_D(Tk,Tl).\] Multiplication by \(S\) is injective, since \(S\) is nonzero on \(D\). Choose decreasing integers \(r_i=v_0(k-2j_i,l-j_i)T^r/r!+O(T^{r-1})\) below their dimensions, and put \(P=r_0\). Starting from the smallest subspace, extend to \(r_i\) independent generators at each stage. Each generator is an individual integral form \(S^b\mathcal M\) of bidegree \((Tk,Tl)\). Its value at a counted point is \((-1)^b\mathcal M(A,B)\), bounded by \[O_T\bigl(N^{h(Tk-2b)+g(Tl-b)}\bigr).\] At least \(r_i\) columns have \(b\ge j_iT\). Summing these exponents, then refining the fixed mesh, gives (76). This argument does not require small coefficients for the rebased form \(S\).

For the local divisibility step, suppose that \(D\) is defined over \(\mathbb Q\) and has an integral model smooth of relative dimension \(r\) at the chosen residue class modulo \(q\). A determinant of \(P\) such common-bidegree columns in that projective residue class has valuation at least \(s_r(P)\), as in (13): normalize by chart coordinates that are units modulo \(q\) and expand in the \(r\) free differences. Combining that divisibility with (76), a prime \(q\ge N^\rho\) forces a proper cut whenever \[ \rho>\frac{1+1/r}{v_0(k,l)^{1/r}} \left\{kh+lg-(2h+g)\int_0^a \frac{v_0(k-2j,l-j)}{v_0(k,l)}\,dj\right\}. \tag{77}\] The strict margin fixes the mesh and determinant degree before \(N\). In particular, for a curve with \(a_1=H\cdot D\), \(b_1=J\cdot D\), the choice \((k,l)=(2,1)\) makes the integral \(1/2\) and the threshold \((2h+g)/(2a_1+b_1)\). This is the projective counterpart of the weighted affine curve estimates in Section 9; the general proof uses those affine estimates throughout.

Analytic relations and box incidence

The main proof uses an exact polynomial relation in each pure direction component. We record a local analytic version with its hypotheses, describe the quintic equality case, and give an elementary box-incidence estimate. These observations explain uses of the cyclotomic geometry; the proof of the density theorem uses the affine constructions of Sections 5 and 6.

The enlarged first quartic cell

The following additional exact checks verify the parameters used in Proposition 26. They use the functions upper and F from the program above. The fixed bidegree ratio is \(t_*=1/2\), for which \((H,H_a,H_b)=(10,5,10)\). In particular, no continuity argument or floating-point computation is needed for the small extension below \(\eta=1\).

epsilon = F(1, 10**6)
etaL, etaH = 1-epsilon, F(3001, 3000)
bL, bH = 4-2*etaH, 2+2*epsilon
t, w = F(1, 2), F(24809, 100000)
m = 3*w
u, v = (etaH-m)/4, (bH-m)/4
assert 0 < epsilon < F(1, 2)
assert 0 < m < 1 and m < min(etaL, bL)
assert w**3*10 > (F(5, 16)*(etaH+bH*t))**4
assert w**2*5 > ((etaH+bH*t)/3)**3
assert min(u, v) > 0 and F(3, 20) <= v/u <= 5
def quartic_delta(h, D):
    return upper((u*v)**h/((2*u+v)**(h-1)*D), h)
P = [quartic_delta(4, u+v), quartic_delta(3, v),
     max(quartic_delta(2, min(3*u, 2*v, u+v)),
         upper(u*u/3, 2)), max(u/2, min(u, v/4))]
assert P == [F(4729, 100000), F(5097, 100000),
             F(6899, 100000), F(76819, 1200000)]
gap = 1-m-sum(max(P[:i]) for i in range(1, 5))
tau = F(1, 50000)
assert gap-4*tau == F(1941, 100000)
assert 1-m-2*u-2*tau == F(15319, 120000)