Shared prime support, small radicals, and square-full polynomial values

Abstract

The radical of an integer records its distinct prime divisors, so an unusually small radical signals that repeated prime factors account for a large part of the integer. We ask how often this phenomenon occurs among the values of a polynomial. For a fixed primitive separable polynomial P in Z[T] of degree d = 2 or 3, with positive leading coefficient, we prove an upper bound of order X^(d/7)Y^(3/7+o(1)) for the number of positive values P(n), with 1 ≤ n ≤ X, whose radical is at most Y, in the stated ranges. The proof isolates the primes shared by the squarefree core and square multiplier of P(n): small shared support is counted by congruences, while large shared support forces a cube divisor and is bounded uniformly using a theorem of Evertse and Silverman. We also determine the exact minimax barrier of the three scalar costs in the method. The same uniform square- and cube-fibre estimates yield, for every fixed separable polynomial P of degree d ≥ 2, an upper bound of order X^(d/5+o(1)) for square-full values. This estimate is nontrivial for d < 5. For quadratics it extends Puchta’s exponent 2/5 from T² + 1 to every fixed separable quadratic polynomial and improves the previously published general unconditional exponent of Wongcharoenbhorn and Meemark.