On Taylor series of zeros of complex-exponent polynomials

TL;DR

This paper derives a new factorization formula for the Taylor series coefficients of zeros of complex-exponent polynomials, extending classical results and connecting to algebraic geometry methods.

Contribution

It introduces a generalized factorization formula for zeros of complex-exponent polynomials and proves related theorems on derivations in commutative rings.

Findings

Provides a new formula for Taylor coefficients of polynomial zeros

Extends classical results to complex-exponent polynomials

Connects to algebraic geometry and hypergeometric series

Abstract

We prove a factorization formula for the Taylor series coefficients of a zero of a polynomial as a function of the polynomial's coefficients. This result extends to more general functions which we call "complex-exponent polynomials". To prove this formula, we prove theorems about derivations on commutative rings. We also show that, when applied to polynomials, our formula recovers the results of Sturmfels obtained with GKZ systems ("Solving algebraic equations in terms of $\mathcal{A}$-hypergeometric series". Discrete Math. 210 (2000) pp. 171-181)