Representing integers by multilinear polynomials
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RESEARCH
Representing integers by multilinear polynomials Albrecht Böttcher1 and Lenny Fukshansky2* * Correspondence:
[email protected] Department of Mathematics, Claremont McKenna College, 850 Columbia Ave, Claremont, CA 91711, USA Full list of author information is available at the end of the article Fukshansky acknowledges support by Simons Foundation Grant #519058. 2
Abstract Let F(x) be a homogeneous polynomial in n ≥ 1 variables of degree 1 ≤ d ≤ n with integer coefficients so that its degree in every variable is equal to 1. We give some sufficient conditions on F to ensure that for every integer b there exists an integer vector a such that F(a) = b. The conditions provided also guarantee that the vector a can be found in a finite number of steps. Keywords: Polynomials, Integer representations, Unimodular matrices, Linear and multilinear forms Mathematics Subject Classification: Primary 11D85, Secondary 11C08, 11C20, 11G50
1 Introduction and main result The problem of finding solutions to a given polynomial equation with integer coefficients goes back to the work of Diophantus, resulting in these equations being called the Diophantine equations. While the linear case likely dates back to Diophantus himself, the quadratic equations have been systematically studied by Gauss and famously led to his composition law for the binary quadratic forms. More generally, Hilbert’s 10th problem, in its contemporary formulation, asks whether there exists an algorithm to decide if a given polynomial equation with integer coefficients has a (nontrivial) integer solution. Matiyasevich’s famous theorem [9] of 1970 (building on previous work of others) gives a negative answer to this problem. In fact, Jones [7] proved in 1980 that the question whether a general Diophantine equation of degree four or larger has a solution in positive integers is already undecidable, and not much else is known for polynomials of degree ≥ 4. One possible approach to Hilbert’s 10th problem is through search bounds. Suppose we can prove that if a polynomial F (x1 , . . . , xn ) with integer coefficients has an integer zero a ∈ Zn \ {0}, then it has one with |a| ≤ C(n, F ), where | · | stands for, say, the supnorm and C(n, F ) is some explicitly given function depending on n and F . Since the set of integer points a ∈ Zn satisfying this condition is finite, we can simply search through all of them checking whether any one of these vectors is a zero of F (x1 , . . . , xn ). This approach will either produce a solution or prove that one does not exist. A survey of known results on search bounds can be found in [8]. While Matiyasevich’s theorem guarantees that search
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bounds are not possible in general, an overview of results of this type in the quadratic case can be found in [5], and the current state of the art for the cubic case is in [2], as well as the references therein. In this note we study a special class of polynomia
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