Semigroup associated with a free polynomial

2019/09/09 by Abbas, Ali, Abdallah Assi, Assi, Abdallah
Computer Science · Mathematics · #05E40 #20M14 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1909.03779

openalex publication_date 2019/09/09 · openalex created_date 2019/09/12 · openalex updated_date 2026/07/28

Abstract

Let \mathbbK be an algebraically closed field of characteristic zero and let \mathbbK_C[[x₁,...,x_e]] be the ring of formal power series in several variables with exponents in a line free cone C. We consider irreducible polynomials f=yⁿ+a₁(\underlinex)y^n-1+…+a_n(\underlinex) in \mathbbK_C[[x₁,...,x_e]][y] whose roots are in \mathbbK_C[[x₁^(1)/(n),...,x_e^(1)/(n)]]. We generalize to these polynomials the theory of Abhyankar-Moh. In particular we associate with any such polynomial its set of characteristic exponents and its semigroup of values. We also prove that the set of values can be obtained using the set of approximate roots. We finally prove that polynomials of \mathbb K[[\underlinex]][y] fit in the above set for a specific line free cone (see Section 4).

Related

• Abhyankar-Moh Semigroups for arbitrary hypersurfaces
• Functional equations in formal power series
• Bases of subalgebras of K[[x]] and K[x]
• On the proportion of irreducible polynomials in unicritically generated semigroups
• About the algebraic closure of formal power series in several variables