Solutions of Polynomial Equations in Subgroups of $$\mathbb {F}_p^*$$

Polynomial Equations in Subgroups and Applications

“…The following result generalises a series of previous estimates of a similar type, see [8,11,13,16,19] and references therein. Theorem 1.2.…”

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We obtain a new bound for the number of solutions to polynomial equations in cosets of multiplicative subgroups in finite fields, which generalises previous results of P. Corvaja and U. Zannier (2013). We also obtain a conditional improvement of recent results of J. Bourgain, A. Gamburd and P. Sarnak (2016) and S. V. Konyagin, S. V. Makarychev, I. E. Shparlinski and I. V. Vyugin (2019) on the structure of solutions to the reduction of the Markoff equation x 2 + y 2 + z 2 = 3xyz modulo a prime p.

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“…• derive a new bound on the number of solutions in subgroups to a systems of several polynomials which covers under a unified setting the results of [8,16,19]; • obtain an improvement of (1.4) under a very plausible conjecture on the number of solutions in subgroups of some particular equation over F * p .…”

“…where P is a bivariate absolutely irreducible polynomial over the finite field F p of p elements and G 1 , G 2 ⊆ F p are multiplicative groups in the algebraic closure of F p , see also [11,13,16,19] for some related results. Motivated by the above results and connections, here we…”