Evertse schlickewei schmidt 2002 linear equations multiplicative group

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theorem_1_1: Evertse, Schlickewei and Schmidt's theorem that a1x1+...+anxn=1 has at most exp((6n)^{3n}(r+1)) nondegenerate solutions in a subgroup of rank r of the n-fold multiplicative group of an algebraically closed field of characteristic 0.

theorem_1_2: Evertse, Schlickewei and Schmidt's theorem that the zero set of a simple linear recurrence of order n at least 3 over an algebraically closed field of characteristic 0 is at most exp((6n)^{3n}) integers and arithmetic progressions in all.

theorem_2_1: Evertse, Schlickewei and Schmidt's theorem that the solutions of y1+...+yn=1 lying close in height to a subgroup of rank r of the n-fold multiplicative group of the algebraic numbers lie in at most exp((5n)^{3n}(r+1)) proper linear subspaces.

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Jan-Hendrik Evertse, Hans Peter Schlickewei, and Wolfgang M. Schmidt, Linear equations in variables which lie in a multiplicative group, Ann. of Math. 155 (2002) 807-836 (author preprint from Evertse's Leiden page; 33 pp.).

For an algebraically closed field K of characteristic 0 (the abstract says any field of characteristic 0), a subgroup Gamma of (K*)^n of finite rank r and a1, ..., an in K*, Theorem 1.1 (p. 2) shows that the equation a1 x1 + ... + an xn = 1 has at most exp((6n)^{3n}(r+1)) nondegenerate solutions x in Gamma, a solution being nondegenerate when no subsum over a nonempty set of indices vanishes. The bound depends only on n and r. Theorem 1.2 (pp. 6-7) deduces that the zero set of a simple linear recurrence of order n >= 3 over such a field is a union of at most exp((6n)^{3n}) integers and arithmetic progressions, and has at most that many elements when the recurrence is nondegenerate. Both follow from Theorem 2.1 (p. 9): for a subgroup Gamma of rank r of the n-fold multiplicative group of the algebraic numbers and n >= 2, the solutions of y1 + ... + yn = 1 of the form x*z with x in Gamma and z of small height relative to x lie in at most exp((5n)^{3n}(r+1)) proper linear subspaces. The proofs use the absolute Subspace Theorem of Evertse and Schlickewei and Schmidt's lower bounds for heights of points on varieties. The paper does not mention Erdős problems.

Source: author preprint. The copy read for this card is the authors' preprint from the Leiden page named above, which states no terms, and it prints no notice; the Annals of Mathematics site footer "Copyright © 2026 Annals of Mathematics" (read 2026-10-02 at https://annals.math.princeton.edu/2002/155-3/p02, the page of another article in the same issue) speaks for the version of record's site, not for this preprint, and the article's DOI 10.2307/3062133 resolves to JSTOR, which was not fetched; the term is unstated.

Read status. Claims checked: Theorems 1.1, 1.2 and 2.1 and their settings were read clause by clause on the page images of the print, and their proofs were followed for structure. Nothing here is independently reviewed.

Bears on.

• #407: the paper does not mention the problem. A representation n = 2^a + 3^b + 2^c 3^d is a nondegenerate solution, all terms being positive, of (1/n)x1 + (1/n)x2 + (1/n)x3 = 1 in a subgroup of (Q*)^3 of rank 4, so Theorem 1.1 bounds the number of representations by a constant independent of n. The problem's site credits the proof that the count is bounded to Evertse, Győry, Stewart and Tijdeman (1988), before this paper.

Results.

• Theorem 1.1 (p. 2): at most exp((6n)^{3n}(r+1)) nondegenerate solutions of a1x1+...+anxn=1 in a subgroup of rank r.
• Theorem 1.2 (pp. 6-7): the zero set of a simple linear recurrence of order n >= 3 is at most exp((6n)^{3n}) integers and arithmetic progressions.
• Theorem 2.1 (p. 9): solutions of y1+...+yn=1 of small height relative to a subgroup of rank r of the algebraic n-fold multiplicative group lie in at most exp((5n)^{3n}(r+1)) proper linear subspaces.

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