2011/09/19 by Adam McCabe, Gregory G. Smith
Mathematics · #Advanced Combinatorial Mathematics #Algebra over a field #Algebraic structures and combinatorial models #Algorithm #Annotation #Artificial intelligence #Commutative Algebra and Its Applications #Computer science #Discrete mathematics #Function (biology) #Infinity #Mathematical analysis #Mathematics #Polynomial #Programming language #Pure mathematics #Rational function #Semantics (computer science) #Series (stratigraphy) #Type (biology) #math.AC #math.AG #math.CO #msc:05E40 #msc:13D40 #msc:52B20
paper · pdf · doi:10.1090/s0002-9939-2012-11808-8
published as Proceedings of the American Mathematical Society 141 (2013) 1883-1892 · 9 pages
arxiv created 2011/09/19 · openalex publication_date 2012/12/20 · arxiv updated 2017/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the linear map sending the numerator of the rational function representing the Hilbert series of a module to that of its <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -th Veronese submodule. We show that the asymptotic behaviour as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> tends to infinity depends on the multidegree of the module and the underlying positively multigraded polynomial ring. More importantly, we give a polyhedral description for the asymptotic polynomial and prove that the coefficients are log-concave.