How many integer homogeneous polynomials at small coprime integers have value of a univariate polynomial?

摘要

Let p(y) = p_my~(m+) p_(m-1)y~(m-1 +···+) p_0 ∈ ?[y] be a polynomial of degree m > 0 in an integer variable. We estimate the number of times it equals some homogeneous polynomial in two variables with integer coefficients, degree at most n, and Euclidean norm at most N evaluated at a pair of small coprime integers (we count this number with the occurring multiplicities). For pairs of coprime integers of absolute value at most H > N/√n, this estimate is γ_(n,p)(H)N~(n+1/m)+O(N~(n+1/m-1)H~3 + N~nH~2), where γ_(n,p)(H) does not depend on N.