Counting integers representable as images of polynomials modulo n

Abstract

Given a polynomial f(x1,x2,…,xt) in t variables with integer coefficients and a positive integer n, let α(n) be the number of integers 0 ≤ a < n such that the polynomialcongruencef(x1,x2,…,xt) ≡ a(modn)issolvable. Wedescribeamethod that allows us to determine the function α associated with polynomials of the form c1xk1+c2xk2+···+ctxkt. Then, we apply this method to polynomials that involve sums and differences of squares, mainly to the polynomials x2 +y2, x2 −y2, and x2 +y2 +z2. © 2019, University of Waterloo. All rights reserved.