Combinatorica

, Volume 34, Issue 6, pp 657–688 | Cite as

A nullstellensatz for sequences over \(\mathbb{F}_p \)

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Abstract

Let p be a prime and let A = (a1,...,a) be a sequence of nonzero elements in \(\mathbb{F}_p \). In this paper, we study the set of all 0–1 solutions to the equation \(a_1 x_1 + \cdots + a_\ell x_\ell = 0\) We prove that whenever p, this set actually characterizes A up to a nonzero multiplicative constant, which is no longer true for <p. The critical case =p is of particular interest. In this context, we prove that whenever =p and A is nonconstant, the above equation has at least p−1 minimal 0–1 solutions, thus refining a theorem of Olson. The subcritical case =p−1 is studied in detail also. Our approach is algebraic in nature and relies on the Combinatorial Nullstellensatz as well as on a Vosper type theorem.

Mathematics Subject Classification (2010)

11D04 11T06 11D45 11P70 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.IMJ, Équipe Combinatoire et OptimisationUniversité Pierre et Marie Curie (Paris 6)ParisFrance

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