Computational Algebra Seminar: Rojas -- On Signs of Trinomials and Hypergeometric Series

Speaker: J. Maurice Rojas (Texas A&M)
Title: On Signs of Trinomials and Hypergeometric Series
Time & Place: 1500 - 1600, Thursday 21 August, SMRI Seminar Rm
Abstract:
A curious question of Koiran is whether one can efficiently decide
the signs of univariate trinomials evaluated on rational numbers. More 
precisely: 
 
                 Given H and D, and a polynomial
                    f(x) := c_1 + c_2 x^d + c_3 x^D 
                 with |c_i|<=H and 0<d<D, and integers p and q 
                 with q nonzero and |p|,|q|<=H, can one determine
                 the sign of f(p/q) in time polynomial in log(DH)?

We give an approach to a positive answer, via hypergeometric series, and
some unusual new Puiseux series that are non-hypergeometric. 
Bell Polynomials, and ChatGPT, make interesting appearance as well. 
This is joint work with Emma Boniface and Weixun Deng.