Rousseau et al.: Divisibility of polynomials and degeneracy of integral points

../

corollary_1_2: For a polynomial g of degree at most 1 over the S-integers that is nonzero at the origin and at the n unit vectors, the S-integral n-tuples for which (1 - x_1 - ... - x_n) x_1 ... x_n divides g(x_1, ..., x_n) are not Zariski-dense in affine n-space; the case g = 1 is the S-unit equation.

lemma_2_4: After blowing up a closed subscheme containing D ∩ W, with D an effective Cartier divisor, W a closed subscheme and D ∩ W of codimension at least 2, a set of points is integral with respect to the strict transform of D outside S exactly when the local height of D is at most that of W outside S, both up to an M_k-constant.

proposition_1_4: In the setting of Theorem 1.3, if D_0 + D_1 + ... + D_r has simple normal crossing singularities then the arithmetically pseudo-hyperbolic variety X minus D is simply connected.

theorem_1_1: For n at least 2 and absolutely irreducible forms F_1, ..., F_r, G of the same degree over the S-integers whose hypersurfaces are in general position, the S-integral points of P^n at which every F_i divides G (r at least 2n+1), or at which the product of the F_i divides G (r at least n+2), are finite outside a closed set Z independent of the number field and of S.

theorem_1_3: For n at least 2, r at least 2n+1 and hypersurfaces D_0, ..., D_r of P^n in general position, the blow-up of P^n along the union of the D_i ∩ D_0, with the strict transforms of D_1, ..., D_r removed, is arithmetically pseudo-hyperbolic.

theorem_1_5: For n at least 2, 2n hyperplanes H_1, ..., H_2n of P^n in general position and points P_i on H_i (i up to n+1) lying on no other H_j, the blow-up of P^n at the P_i minus the strict transform of H_1 + ... + H_2n is arithmetically pseudo-hyperbolic.

theorem_1_6: For n at least 2 and q at least 3n hyperplanes H_i of P^n in general position, indexed cyclically, blowing up the q points P_i where H_i, ..., H_(i+n-1) meet and removing the strict transforms of all H_i leaves an arithmetically pseudo-hyperbolic variety.

theorem_1_7: For n at least 2 and q at least 3n linear forms F_i in general position, indexed cyclically, the k-points of P^n at which, for every i, the ideal F_i(x)(x_0, ..., x_n) equals the product of the ideals generated by n consecutive forms lie in a Zariski closed subset Z of P^n.

theorem_4_1: On a Cohen-Macaulay projective n-fold with divisors D_0, ..., D_r in general position, D_i numerically d_i A for one ample A and d_i at least d_0, the k-points where every normalized local height of D_i is at most that of D_0 outside S (r at least 2n+1), or where their sum is (r at least n+2), are finite outside a proper closed set independent of k, S and the M_k-constant.

---

The copy read for this card is the arXiv version stamped "arXiv:2106.11337v1 [math.NT] 21 Jun 2021", the edition this card cites. Provenance: downloaded from https://arxiv.org/pdf/2106.11337v1 on 2026-09-25; 354,822 bytes. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2106.11337), every other right reserved.

Erwan Rousseau, Amos Turchet, Julie Tzu-Yueh Wang, "Divisibility of polynomials and degeneracy of integral points," Math. Ann. 388 (2024), no. 2, 1969--1999, doi:10.1007/s00208-023-02564-3; preprint arXiv:2106.11337 (2021). Theorem, lemma and equation numbers on this card are those of arXiv v1.

Bears on: Problem 699, as a framework only: the minimum of local heights in (2.1) and (2.3) is, for a suitable point and two coordinate lines, the quantity whose vanishing at every prime $p\ge I$ characterizes a counterexample, and Lemma 2.4 turns a comparison of such local heights into integrality on a blow-up; but no theorem of the paper applies to the problem's binomial coefficients, and the paper proves or excludes no case of it (see Relation to E699).

Read status. Claims checked for Theorems 1.1, 1.3, 1.5, 1.6, 1.7 and 4.1, Corollary 1.2, Proposition 1.4, Lemma 2.4 and Definitions 2.2 and 2.3, read clause by clause against the page images of arXiv v1; the proofs were read for their structure only and not re-derived.

Result pages.

• Theorem 1.1 (p. 2): divisibility of polynomial values at $S$-integral points is degenerate.
• Corollary 1.2 (p. 2): the higher-dimensional $S$-unit-type divisibility.
• Theorem 1.3 (p. 2): the blow-up along the $D_i\cap D_0$, with Theorem 4.2.
• Proposition 1.4 (p. 2): simple connectivity of that complement.
• Theorem 1.5 (p. 3): $2n$ hyperplanes and $n+1$ blown-up points.
• Theorem 1.6 (p. 3): the cyclic configuration of $q\ge3n$ hyperplanes.
• Theorem 1.7 (p. 3): the cyclic system of ideal equalities.
• Lemma 2.4 (p. 6): integrality on a blow-up as a local-height inequality.
• Theorem 4.1 (p. 8): the general local-height finiteness theorem.

Overview

The paper studies when divisibility relations among values of several fixed polynomials force the corresponding rational points into a proper algebraic subset. Its geometric formulation concerns integral points on blow-ups of projective varieties, with the principal goal of constructing simply connected quasi-projective varieties whose integral points are arithmetically degenerate.

The basic arithmetic result is Theorem 1.1. For $n\ge2$ and absolutely irreducible homogeneous polynomials $F_1,\ldots,F_r,G\in\mathcal O_S[x_0,\ldots,x_n]$ of the same degree, with $\deg F_i\ge\deg G$, defining hypersurfaces in general position, it gives a closed set $Z\subset\mathbf P^n$, independent of the number field and $S$, outside which only finitely many $S$-integral points satisfy either all divisibilities $F_i(x)\mid G(x)$ when $r\ge 2n+1$, or the product divisibility $\prod_iF_i(x)\mid G(x)$ when $r\ge n+2$. Corollary 1.2 specializes the product case to the linear factors $x_1,\ldots,x_n,1-\sum x_i$, obtaining non-density for a higher-dimensional $S$-unit-type equation.

The more general engine is Theorem 4.1. On a Cohen–Macaulay projective $n$-fold, divisors $D_0,\ldots,D_r$ in general position with $D_i\equiv d_iA$ and $d_i\ge d_0$ satisfy the same finiteness conclusion under normalized local-height inequalities $\frac1{d_i}\lambda_{D_i,v}\leq\frac1{d_0}\lambda_{D_0,v}+\gamma_v$ or $\sum_i\frac1{d_i}\lambda_{D_i,v}\leq\frac1{d_0}\lambda_{D_0,v}+\gamma_v$. The polynomial statement follows by converting divisibility into these inequalities in the proof of Theorem 1.1. Equations (4.1)–(4.2) give the normalized conditions, while inequalities (4.4)–(4.11) compare counting functions and heights and yield bounded height outside the exceptional set.

The divisibility–geometry dictionary is isolated in Lemma 2.4. If $\widetilde X\to X$ blows up a center containing $D\cap W$, integrality relative to the strict transform of $D$ is equivalent to $\lambda_{D,v}\leq\lambda_{W,v}$ away from $S$. This follows from the pullback identity (2.2) and the intersection formula $\lambda_{D\cap W,v}=\min\{\lambda_{D,v},\lambda_{W,v}\}$ in (2.3), itself based on Definition–Theorem 2.1 and equation (2.1). Theorem 4.2 applies this dictionary to blow-ups of the unions $D_i\cap D_0$; Theorem 1.3 is its specialization to hypersurfaces in $\mathbf P^n$. Proposition 1.4 adds that the resulting complement is simply connected when $D_0+\cdots+D_r$ has simple normal crossings.

The main Diophantine input is the Ru–Vojta inequality, stated as Theorem 3.4 and equation (3.1), with an exceptional set independent of the field and $S$. Its coefficient $\beta_{\mathcal L,D}$ is defined in Definition 3.1. The proof of Theorem 4.1 blows up $Y=\bigcup_i(D_i\cap D_0)$, uses Proposition 4.7 to retain general position of the pullbacks, and applies the estimate for $\beta^{-1}$ in Lemma 4.6 and (4.3). The field-independent exceptional set is traced through the refinement of Schmidt’s subspace theorem in Theorem 3.5, equation (3.2). Proposition 4.3, Theorem 4.4, and Lemma 3.6 supply the required global/local height comparisons.

Sections 5 and 6 treat two further blow-up configurations. Corollary 5.5 gives a lower bound for $\beta_{A,B}$ in terms of intersection numbers, derived from the section estimate (5.1) of Lemma 5.4. Lemma 5.6 computes the asymptotic bounds (5.4)–(5.5), leading to Theorem 1.5: the complement of the strict transforms of $2n$ suitably positioned hyperplanes after blowing up $n+1$ points is arithmetically pseudo-hyperbolic. For the cyclic configuration of $q\ge3n$ hyperplanes, Lemma 6.1 proves nefness of $D-m\widetilde H_i$ for $0\le m\le n$, and Lemma 6.2 uses (6.9)–(6.14) to prove that $D$ is big and $\beta_{D,\widetilde H_i}>1$. This yields arithmetic pseudo-hyperbolicity in Theorem 1.6. Theorem 1.7 translates it into containment in a Zariski closed set (not stated to be proper) for points satisfying the system of ideal equalities (6.17); equations (6.18)–(6.20) are the local-height translation.

The conclusions are qualitative: outside a proper closed subset, relevant integral sets are finite (Definition 2.3), but the exceptional subset and finite residual set are not explicitly determined. Sections 7 and 8 establish parallel analytic and function-field results. Theorem 7.1 is the entire-function divisibility statement, concluding algebraic dependence, with thresholds $r\ge2n$ and $r\ge n+1$ in place of $2n+1$ and $n+2$; Theorems 7.4–7.6 give Brody pseudo-hyperbolicity, using the analytic Ru–Vojta theorem (Theorem 7.8). Theorems 8.2–8.4 give algebraic pseudo-hyperbolicity over characteristic-zero function fields, based on Theorem 8.5. These are proved analogously and are not additional number-field divisibility theorems.

Relation to E699

Write the variables of E699 as $(N,I,J)$, with $1\le I<J\le N/2$, and put $b_r(N)=\binom Nr$. Let $S_I=\{\infty\}\cup\{p:p<I\}$. The desired assertion is exactly

$$\sum_{p\notin S_I}\min\{v_p(b_I(N)),v_p(b_J(N))\}\log p>0.$$

Thus a counterexample is characterized by the vanishing of this sum, equivalently by $\gcd(b_I(N),b_J(N))$ being supported only on primes $<I$.

This is the kind of local expression encoded by Definition–Theorem 2.1: the local height of an intersection of divisors is the minimum of their local heights, as in (2.1) and (2.3). For the point $[b_I(N):b_J(N):1]\in\mathbf P^2$ and the coordinate lines $x_0=0$ and $x_1=0$ with their standard Weil functions, the displayed sum is the outside-$S_I$ counting function of the intersection of the two lines at that point, the logarithm of the part of the gcd supported on primes $\ge I$. Lemma 2.4, whose proof rests on (2.3), supplies the geometric mechanism by which comparisons of such valuations become integrality conditions on a blow-up. This dictionary is the paper’s principal usable contribution to E699: it suggests representing hypothetical counterexamples as integral points on an appropriate blow-up and then seeking degeneracy or finiteness.

The paper’s theorems do not directly apply, however. For fixed $I$ and $J$, $\binom XI$ and $\binom XJ$ are one-variable, reducible, rational-coefficient polynomials with nested linear factors; their divisors therefore fail the absolute-irreducibility and general-position hypotheses of Theorem 1.1. Clearing the factorial denominators changes valuations at primes between $I$ and $J$, precisely within E699’s relevant range. Moreover, E699 imposes a small-gcd condition on a counterexample, whereas Theorems 1.1 and 4.1 treat inequalities expressing divisibility of many polynomial values by another value. Their numerical thresholds require, besides the divisor $D_0$, at least $2d+1$ divisors $D_i$, or $d+2$ in the product case, on a fixed $d$-dimensional variety; E699 supplies only two binomial values and lets both $I$ and $J$ vary.

The ideal identities in Theorem 1.7 and (6.17) are structurally closer to gcd data, but they require a cyclic collection of $q\ge3d$ linear forms in general position and simultaneous exact identities. They imply only containment in an unspecified Zariski-closed set, which the statement does not call proper, not the existence of a prime divisor above a moving threshold. Likewise, arithmetic pseudo-hyperbolicity leaves an exceptional subvariety and finitely many points untreated and supplies no effective bounds.

Accordingly, the source neither proves E699 nor excludes any of its remaining indices. It is relevant as a conceptual blow-up/local-height framework: if the binomial conditions could be embedded into a fixed higher-dimensional, general-position divisor configuration, Theorem 4.1 or the $\beta>1$ strategy of Lemma 6.2 could potentially force a counterexample family into a proper exceptional locus. Substantial new work would still be needed to construct such a configuration, analyze that locus, preserve the exact cutoff $p\ge I$, and handle all individual triples.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.