Products with variables from low-dimensional affine spaces and shifted power identity testing in finite fields

Motivated by some algorithmic applications, we obtain upper bounds on the number of solutions of the equation x1…xn=λx1…xn=λ with variables x1,…,xnx1,…,xn from a low-dimensional affine space in a high degree extension of a finite field. These are analogues of several recent bounds on the number of solutions of congruences of the similar form with variables in short intervals. We apply this to the recently introduced algorithmic problem of identity testing between shifted power functions in finite fields.