On the maximal number of roots of a trinomial over a prime field

Canetti, Friedlander, et al. (2002) studied the randomness of powers over finite fields and along the way derived an analogue of Descartes’™ rule over the finite field F_q with q elements: They showed that the number of roots of any univariate t-nomial, with exponents {0,a_2,...,a_t} and the differences a_i-a_j all relatively prime to q-1, is O(q^{(t-2)/(t-1)}). The correct optimal bounds remain a mystery for prime fields, even in the case of polynomials with three terms. Following the work of Kelley (2016), we seek to prove the conjecture that the number of roots in F_p of a trinomial with a linear middle term is always O(log p). We expand current evidence by using a supercomputer to determine the  number of roots of these trinomials for 139,571 < p 191,491. We also prove that the search can be restricted to trinomials with a middle linear term when p-1 has less than three distinct prime factors.

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