ADDITION THEOREMS IN Fp VIA THE POLYNOMIAL METHOD

Abstract : In this article, we use the Combinatorial Nullstellensatz to give new proofs of the Cauchy-Davenport, the Dias da Silva-Hamidoune and to generalize a previous addition theorem of the author. Precisely, this last result proves that for a set A ⊂ Fp such that A ∩ (−A) = ∅ the cardinality of the set of subsums of at least α pairwise distinct elements of A is: |Σα(A)| ≥ min (p, |A|(|A| + 1)/2 − α(α + 1)/2 + 1) , the only cases previously known were α ∈ {0, 1}. The Combinatorial Nullstellensatz is used, for the first time, in a direct and in a reverse way. The direct (and usual) way states that if some coefficient of a polynomial is non zero then there is a solution or a contradiction. The reverse way relies on the coefficient formula (equivalent to the Combinatorial Nullstellensatz). This formula gives an expression for the coefficient as a sum over any cartesian product. For these three addition theorems, some arithmetical progressions (that reach the bounds) will allow to consider cartesian products such that the coefficient formula is a sum all of whose terms are zero but exactly one. Thus we can conclude the proofs without computing the appropriate coefficients.
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Submitted on : Thursday, February 16, 2017 - 7:53:06 PM
Last modification on : Sunday, March 31, 2019 - 1:28:40 AM
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  • HAL Id : hal-01469950, version 1
  • ARXIV : 1702.06419

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Eric Balandraud. ADDITION THEOREMS IN Fp VIA THE POLYNOMIAL METHOD. 2017. ⟨hal-01469950⟩

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