Conditional expanding bounds for two-variable functions over arbitrary fields.

In this paper, we prove some results on the sum-product problem over arbitrary fields which improve and generalize results given by Hegyvári and Hennecart [5] . More precisely, we prove that, for related pairs of two-variable functions f ( x , y ) and g ( x , y ) , if A and B are two sets in an arbitrary field F with | A | = | B | , then max ⁡ { | f ( A , B ) | , | g ( A , B ) | } ≫ | A | 1 + c , for some c > 0 . [ABSTRACT FROM AUTHOR]