On the maximal number of roots of a trinomial over a prime field
MSRIUP 2017: Solving Systems of Polynomial Equations June 24, 2017  August 06, 2017
Location: SLMath: Baker Board Room
Dastidar, Márquez, Pugh
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 tnomial, with exponents {0,a_2,...,a_t} and the differences a_ia_j all relatively prime to q1, is O(q^{(t2)/(t1)}). 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 p1 has less than three distinct prime factors.
Dastidar, Márquez, Pugh
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