For the quantum integer ½n q ¼ 1 þ q þ q 2 þ ? þ q nÀ1 there is a natural polynomial multiplication such that ½m q # q ½n q ¼ ½mn q : This multiplication leads to the functional equation f m ðqÞf n ðq m Þ ¼ f mn ðqÞ; defined on a given sequence F ¼ f f n ðqÞg N n¼1 of polynomials. This paper contains various results concerning the construction and classification of polynomial sequences that satisfy the functional equation, as well open problems that arise from the functional equation. r
The Erdős-Heilbronn conjectureThe Cauchy-Davenport theorem states that if A and B are nonempty sets of congruence classes modulo a prime p, and if |A| = k and |B| = l, then the sumset A + B contains at least min(p, k + l − 1) congruence classes. It follows that the sumset 2A contains at least min(p, 2k − 1) congruence classes. Erdős and Heilbronn conjectured 30 years ago that there are at least min(p, 2k − 3) congruence classes that can be written as the sum of two distinct elements of A. Erdős has frequently mentioned this problem in his lectures and papers (for example, Erdős-Graham [4, p. 95]). The conjecture was recently proven by Dias da Silva and Hamidoune [3], using linear algebra and the representation theory of the symmetric group. The purpose of this paper is to give a simple proof of the Erdős-Heilbronn conjecture that uses only the most elementary properties of polynomials. The method, in fact, yields generalizations of both the Erdős-Heilbronn conjecture and the Cauchy-Davenport theorem.
The polynomial methodLemma 1 (Alon-Tarsi [2]) Let A and B be nonempty subsets of a field F with |A| = k and |B| = l. Let f (x, y) be a polynomial with coefficients in F and
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