EXPANDING POLYNOMIALS ON SETS WITH FEW PRODUCTS. Issue 1 (26th November 2019)

• Record Type: Journal Article
• Authors: Pohoata, Cosmin
• Is Part Of: Mathematika. Volume 66:Issue 1(2020)
• Page Start: 71
• Page End: 78
• Publication Date: 2019-11-26
• Subjects: 11B13 -- 11J87 (primary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/mtk.12007
• Languages: English
• ISSNs: 0025-5793

Abstract

Abstract: In this note, we prove that if A is a finite set of real numbers such that | A A | = K | A |, then for every polynomial f ∈ R [ x, y ], we have that | f ( A, A ) | = Ω K, deg f ( | A | 2 ), unless f is of the form f ( x, y ) = g ( M ( x, y ) ) for some monomial M and some univariate polynomial g . This is sharp up to the dependence on K and the degree of f .