Roots of Polynomials in Subgroups of $\mathbb F^*_p$ and Applications to Congruences

The study of congruences (mod   p) for sparse polynomials in one variable arises in several contexts and an important special case occurs when differences between exponents of the monomials have large common factors with p−1. In this situation, usual techniques based on Fourier analysis (mod    p) and Weil's bounds for exponential sums do not work well and new methods are needed. In this paper, we introduce a new technique based on a new result (Proposition 1) that shows in a precise quantitative way that it is not possible to describe a “large” subgroup of $\mathbb F^*_p$ using only polynomial equations of small degree and small height; such a result may be viewed as yet another example of the “independence” between addition and multiplication in a finite field. Besides showing in Theorem 17 that the number of solutions of the congruences studied here admits a lower bound of the expected order in “small boxes,” an example is given in which the actual number of solutions is much larger than what can be expected from a probabilistic counting. The reduction of the original problem to the principle alluded to above involves the geometry of intersections of varieties of Fermat-type, arithmetic formulations of Bézout's Theorem and of Hilbert's Nullstellensatz, as well as certain higher-dimensional versions of Proposition 1.