Distribution of integer points on determinant surfaces and a mod‐ <i>p</i> analogue

Abstract We establish an asymptotic formula for counting integer solutions with smooth weights to an equation of the form , where is a non‐zero integer, with an explicit main term and a strong bound on the error term in terms of the size of the variables as well as of . We also establish an asymptotic formula for counting integer solutions with smooth weights to the congruence , where is a large prime, with a strong bound on the error term.