On Taylor series of zeros of complex-exponent polynomials

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)