D'Andrea, Carlos, 1973-Quingles i Davı́, Guillem2022-05-132022-05-132021-06-20https://hdl.handle.net/2445/185526Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2021, Director: Carlos D'Andrea[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.54 p.application/pdfcatcc-by-nc-nd (c) Guillem Quingles i Davı́, 2021http://creativecommons.org/licenses/by-nc-nd/3.0/es/Combinatòria (Matemàtica)Treballs de fi de grauFuncions de diverses variables complexesFuncions holomorfesCombinationsBachelor's thesesFunctions of several complex variablesHolomorphic functionsinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/openAccess