Corvaja and Zannier (2011): an abcd theorem over function fields

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corollary_p441: Corvaja and Zannier's corollary, stated without proof: for each epsilon > 0 there is delta(epsilon, g) > 0 such that multiplicatively independent S-units u, v with #(S) at most delta H* satisfy H* < (1 + epsilon) #(S_z).

recalled_abc_abcd_bounds: States the three- and four-summand function-field height bounds recalled on page 438, including support, subsum, and normalized nonconstancy conditions.

theorem_1_1: Corvaja and Zannier's abcd theorem: for S-units u, v on a curve, not both constant, with z = u + v + 1 not 0, 1, u or v, the number of zeros of z outside S is at least the height of (1:u:v) minus explicit error terms in chi, with a separate bound when u and v are multiplicatively dependent.

theorem_1_1_star: The alternative form of the first case of Corvaja and Zannier's Theorem 1.1: for multiplicatively independent S-units u, v with z = u + v + 1 not 0, 1, u or v, the cube root of H* is at most the cube root of #(S_z) + 16 chi plus the cube root of 2^14 chi.

theorem_1_2: Corvaja and Zannier's theorem that for 1/a + 1/b < 2.5 x 10^-4 and nonconstant rational functions x(t), y(t), a nonconstant perfect power x^a + y^b + 1 must be a square, with a = 2b and y^2b = 4x^a or b = 2a and 4y^b = x^2a.

theorem_1_2_bis: Corvaja and Zannier's theorem that for c >= 2 and a >= 10^4 the affine surface x^a + y^a + z^c = 1 contains only finitely many curves of geometric genus at most 1, a case of Bogomolov's conjecture.

theorem_1_3: Corvaja and Zannier's theorem that three distinct nonzero complex polynomials, not all constant, whose pairwise products plus 1 are perfect powers with exponents at least 864 must, after permutation, satisfy c^2 + 1 = 0 and a + b = 2c.

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Pietro Corvaja and Umberto Zannier, An abcd theorem over function fields and applications, Bulletin de la Société Mathématique de France 139 (2011), no. 4, 437-454. doi:10.24033/bsmf.2613. The copy read for this card is the published journal PDF, available from Numdam, BSMF_2011__139_4_437_0. The file prints "© Société Mathématique de France" on printed p. 437 (PDF p. 2), every other right reserved.

The PDF has 19 pages including a cover: PDF page 2 is printed page 437, PDF page 3 is printed page 438, and PDF page 19 is printed page 454. The article records receipt on 9 April 2008, revisions on 14 September 2009 and 1 July 2010, and acceptance on 24 September 2010. The copy read is the journal publication itself; its PDF-generation metadata marks no separate mathematical revision.

Results

Section 1 (pp. 438-442) recalls the setting: $\kappa$ algebraically closed of characteristic zero, a smooth complete curve of genus $g$, a finite set $S$ of its points with $\#(S)\ge2$, $\chi=2g-2+\#(S)$, and the projective height $H$. For $S$-units $u,v$ and $z=u+v+1\ne0,1,u,v$ it puts $\tilde H=H(1:u:v)$ and $H^*=\tilde H+\chi+\#S$.

• Recalled abc and abcd bounds (p. 438, unnumbered): the Mason-Stothers bound $H(1:u:v)\le\chi$ when $1+u+v=0$ and $u,v$ are not both constant, and the four-summand Brownawell-Masser bound $H(1:u:v:z)\le3\chi$ when no subsum vanishes. They are recalled external results, not results of this paper.
• Theorem 1.1 (p. 440), the paper's main result: for $S$-units $u,v$, not both constant, the number of zeros of $z$ outside $S$ is at least $\tilde H-15\chi-6H^{*2/3}\chi^{1/3}$ when $u,v$ are multiplicatively independent modulo $\kappa^*$, and at least $\tilde H(1-1/\max(|r|,|s|))-15\chi$ under a relation $u^r=\lambda v^s$ with $r,s$ coprime.
• Theorem 1.1* (p. 441): the independent case recast as an upper bound for $H^*$ in terms of $\#(S_z)$ and $\chi$.
• Corollary (p. 441, unnumbered, given without proof): $H^*<(1+\epsilon)\#(S_z)$ when $\#(S)\le\delta(\epsilon,g)H^*$ and $u,v$ are multiplicatively independent, the coefficient that Vojta's conjecture predicts.
• Theorem 1.2 (p. 441): for $1/a+1/b<2.5\cdot10^{-4}$ and nonconstant $x,y\in\kappa(t)$, a nonconstant perfect power $x^a+y^b+1$ is a square, with $a=2b$ and $y^{2b}=4x^a$ or $b=2a$ and $4y^b=x^{2a}$.
• Theorem 1.2 bis (p. 441): for $c\ge2$ and $a\ge10^4$ the surface $x^a+y^a+z^c=1$ contains only finitely many affine curves of geometric genus at most $1$, which the paper presents as a case of Bogomolov's conjecture.
• Theorem 1.3 (p. 442): three distinct nonzero complex polynomials, not all constant, with $1+ab$, $1+ac$, $1+bc$ perfect powers of exponents at least $864$ satisfy $c^2+1=0$ and $a+b=2c$ after permutation.

The proofs (Section 2, pp. 442-449) rest on Lemma 2.1 (p. 443), a height bound the paper derives from Theorem 1 of U. Zannier, Some remarks on the $S$-unit equation in function fields, Acta Arith. 64 (1993), 87-98; Lemma 2.2 (p. 443) on zeros of differentials; and Theorem CZ (pp. 443-444), a bound for the common zeros of $u-1$ and $v-1$ taken from Corollary 2.3 of the authors' Some cases of Vojta's conjecture on integral points over function fields, J. Algebraic Geom. 17 (2008), 295-333. The Appendix (pp. 449-454) proves Proposition A (p. 450): the surface $x^a+y^a+z^c=1$ is of general type when $c\ge2$ and $a>3c/(c-1)$, the input that Theorem 1.2 bis needs in genus one.

Attribution of the recalled bounds

The source attributes the three-summand bound to Mason and Stothers, and the four-summand bound to the inequalities of Brownawell and Masser. Its bibliography, p. 454, identifies the latter source as W. D. Brownawell and D. W. Masser, Vanishing sums in function fields, Mathematical Proceedings of the Cambridge Philosophical Society 100 (1986), 427-434. No theorem label from that original paper has been checked here. The abstract also credits Voloch in its historical description of the abcd theorem. The Browkin-Brzezinski attribution on p. 438 concerns a sharpness example, not authorship of the height inequality.

Reading coverage

The cover and printed pages 437-454 were read on the page images. The statements, hypotheses and constants of the results listed above were checked clause by clause; the proofs were read but not checked step by step, and the Corollary on p. 441 has no printed proof. These are claims-checked interfaces. No proof, classical or of this paper, has been reconstructed or independently accepted here.

Bears on.

• Problem 477: the recalled height bounds are possible inputs to excluding polynomial families in translate equations. Applying them requires verification of the normalized functions, their zero and pole support, and the nondegeneracy conditions; this card does not review that application or an additive-complement conclusion.
• Problem 939: the Problem 939 research pages use the recalled three-summand bound in genus zero and note that the recalled four-summand bound does not exclude the four-term identities of $5$-powerful polynomials that they seek. The recalled bounds decide no part of Problem 939, and the paper's own theorems are not used there.

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