[Paper Review] Addition Theorems in Fp via the Polynomial Method

Éric Balandraud|arXiv (Cornell University)|Feb 21, 2017

Mathematics and Applications3 citations

TL;DR

This paper uses the Combinatorial Nullstellensatz and its coefficient formula in a novel reverse application to prove a generalized addition theorem in finite fields 𝔽_p. It establishes a lower bound on the size of the set of subsums of at least α distinct elements from a subset A ⊂ 𝔽_p with A ∩ (−A) = ∅, showing |Σ_α(A)| ≥ min{p, |A|(|A|+1)/2 − α(α+1)/2 + 1}, extending prior results for α = 0,1 to all α ∈ [0, |A|].

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 $\subset$ Fp such that A $\cap$ (--A) = $\emptyset$ the cardinality of the set of subsums of at least $α$ pairwise distinct elements of A is: |$Σ$$α$(A)| $\ge$ min (p, |A|(|A| + 1)/2 -- $α$($α$ + 1)/2 + 1) , the only cases previously known were $α$ $\in$ {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.

Motivation & Objective

• To extend the known addition theorems in 𝔽_p to subsums of at least α distinct elements, beyond the previously known cases α ∈ {0,1}.
• To apply the Combinatorial Nullstellensatz in a reverse manner via the coefficient formula, enabling direct proof without explicit coefficient computation.
• To establish sharp lower bounds on the size of Σ_α(A), the set of subsums of at least α pairwise distinct elements of A, under the condition A ∩ (−A) = ∅.
• To unify and generalize results on subsum sets using symmetric and arithmetic progression constructions that achieve the bounds.

Proposed method

• Applies the Combinatorial Nullstellensatz in a reverse way: instead of showing a coefficient is non-zero to derive a solution, uses the coefficient formula to express a coefficient as a sum over a Cartesian product.
• Constructs a multivariate polynomial whose coefficient encodes the number of solutions to subsum problems with distinct elements.
• Uses the coefficient formula to show that exactly one term in the sum is non-zero when the set A is an arithmetic progression, implying the coefficient is non-zero.
• Employs symmetric properties of subsum sets: Σ_α(A) = (ΣA) − Σ^α(A), which implies |Σ_α(A)| = |Σ^α(A)|.
• Analyzes binomial and double factorial expressions to evaluate the coefficient in the polynomial, particularly for extremal cases like A = [1,d].
• Relies on combinatorial identities involving binomial determinants and ballot numbers to validate the coefficient's non-vanishing nature in critical configurations.

Experimental results

Research questions

• RQ1 What is the minimal possible size of the set of subsums of at least α distinct elements from a subset A ⊂ 𝔽_p with A ∩ (−A) = ∅?
• RQ2 Can the Combinatorial Nullstellensatz be applied in a reverse manner to prove lower bounds on subsum sets without computing coefficients explicitly?
• RQ3 How does the size of Σ_α(A) depend on α and |A|, and what configurations achieve the minimal size?
• RQ4 Is the bound |Σ_α(A)| ≥ min{p, |A|(|A|+1)/2 − α(α+1)/2 + 1} sharp for all α ∈ [0, |A|]?
• RQ5 Can the method used for α = 0,1 be generalized to arbitrary α using symmetric and extremal constructions?

Key findings

• The paper proves that for any A ⊂ 𝔽_p with A ∩ (−A) = ∅ and any α ∈ [0, |A|], the size of the set of subsums of at least α distinct elements satisfies |Σ_α(A)| ≥ min{p, |A|(|A|+1)/2 − α(α+1)/2 + 1}.
• The bound is sharp, as equality is achieved when A is the arithmetic progression [1, |A|], yielding Σ^α(A) = [0, |A|(|A|+1)/2 − α(α+1)/2], which has exactly min{p, |A|(|A|+1)/2 − α(α+1)/2 + 1} elements.
• The proof technique is novel in its use of the coefficient formula of the Combinatorial Nullstellensatz in reverse: by choosing a Cartesian product where all but one term vanish, the non-vanishing term implies the coefficient is non-zero.
• For α = 0 and α = 1, the result reduces to the previously known Theorem 3, thus generalizing it to all α ∈ [0, |A|].
• The method avoids explicit coefficient computation by leveraging symmetric properties and extremal configurations (e.g., arithmetic progressions) to isolate a single non-zero term in the coefficient sum.
• The approach demonstrates that the polynomial method, when combined with the coefficient formula, can yield tight bounds in additive combinatorics even in complex subsum problems.