Extremal Trinomials over Quadratic Finite Fields

In the process of pursuing a finite field analogue of Descartes’ Rule, Bi, Cheng, and Rojas (2014) proved an upper bound of 2 √ q − 1 on the number of roots of a trinomial c1 + c2x a2 + c3x a3 ∈ Fq [x], conditional on the exponents satisfying δ = gcd(a2, a3, q − 1) = 1, and Cheng, Gao, Rojas, and Wan (2015) showed that this bound is near-optimal for many cases. Our project set out to refine these results by finding new bounds and extremal examples for cases not yet explored. We construct a class of extremal examples having √ q roots on fields with q an even power of a prime, by using linear maps with large null spaces over Fq. Additionally, we present a new upper bound of 1+ √ 4q−7 2 for all q under the same constraints as before, using an alternate method involving reduced trinomials with disjoint root sets, which establishes the examples as maximal. Our methods offer several possible generalizations to other cases. 1 Background and Objectives Sparse polynomials, which have a fixed number of terms, regardless of their degree, often have much lower maximum numbers of roots than general polynomials over the same field. For example, over the real numbers, Descartes’s Rule states that a polynomial with t nonzero roots must have less than 2t real roots. This bound is sharp; for example, the polynomial x(x−1)(x−2) · · · (x−(t−1)) has t terms and 2t− 1 roots [2]. Our current work is part of an effort to formulate an analogous rule for finite fields Fq. We define a univariate t-nomial as f(x) = c1 + c2x2 + · · · + ctxt with a2 < · · · < at < q − 1, and for such f we define δ = gcd(a2, . . . , at, q − 1). Bi, Cheng and Rojas (2014) recently showed that the nonzero roots of t-nomials over Fq display a peculiar multiplicative structure: they can be partitioned into cosets of two subgroups of Fq , whose number and size are predictable based on q, t, and δ. Theorem 1.1 (Thm. 1.1 in [1]). The nonzero roots of a univariate t-nomial as defined above are the union of at most 2 ( q−1 δ ) t−2 t−1 cosets of two subgroups S1 ⊆ S2, with |S1| = δ and |S2| ≥ δ ( q−1 δ ) 1 t−1 .