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


Let p be a prime and let A = (a 1,...,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.

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Corresponding authors

Correspondence to Éric Balandraud or Benjamin Girard.

Additional information

Research supported by the French ANR Project “CAESAR” No. ANR-12-BS01-0011

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Balandraud, É., Girard, B. A nullstellensatz for sequences over \(\mathbb{F}_p \) . Combinatorica 34, 657–688 (2014). https://doi.org/10.1007/s00493-011-2961-4

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Mathematics Subject Classification (2010)

  • 11D04
  • 11T06
  • 11D45
  • 11P70