Sèries de Taylor de zeros de polinomis d’exponents complexos

Abstract

[en] We follow the work of DeFranco in [4] and [5] to 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 and also to the sum of a complex exponent polynomial and an holomorphic function with a simple zero. To prove this formula we need lemmas about Stirling numbers, multisets and partition sets. We also show that, when applied to polynomials, the formula recovers Sturmfels results in [11]. Finally, continuing the the work of DeFranco, we see that the formula, when applied to second degree polynomials, agrees with the known radical solutions and we prove an extension of a result about derivations on commutative rings.