The Taxicab Problem for Polynomials and Generalizations of Mason's Theorem

Abstract: This talk is motivated by Ramanujan's famous taxicab problem and is concerned with the solvability of polynomial equations of the form $p^n+q^n=r^n+s^n$ and, more generally, $p_1^{k_1}+\dots+p_m^{k_m}=0$ over the complex numbers. Using Wronskian determinants and Mason's theorem, we obtain sharp upper bounds for the exponents. In particular, we will show that there are no relatively prime polynomials (with at least one non-constant) satisfying the generalised taxicab equation for $n \ge 10$. We also consider an extension of Mason's theorem to $f_0+f_1+\dots+f_k=0$ for several polynomial terms over the complex numbers and finite fields, obtaining the corresponding degree bounds. Finally, the talk points out interesting future cryptographic applications of these theoretical results, in particular, the construction of large families of pseudorandom binary sequences with small cross-correlation measures.

number theory

Audience: researchers in the topic

Binghamton Arithmetic Seminar