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Mathematics   > Home   > Advances in Pure and Applied Mathematics   > Issue 2 (March 2025)   > Article

On the index of regularity of additive decompositions of forms

Sur l’index de régularité des décompositions additives des formes


Edoardo Ballico
University of Trento
Italy



Published on 20 March 2025   DOI : 10.21494/ISTE.OP.2025.1259

Abstract

Résumé

Keywords

Mots-clés

Let $$$f$$$ be a degree $$$d$$$ form in $$$n+1$$$ variables $$$x_0,\dots ,x_n$$$. Any additive decomposition of $$$f$$$ is associated to a finite set $$$A\subset ℙ^n$$$ with $$$\#A$$$ the number of non-proportional addenda. We study the index of regularity $$$\rho(A)$$$ of $$$A$$$, i.e. the first integer $$$t$$$ such that $$$h^1(\mathcal{I}_A(t)) = 0$$$, of the finite subset $$$A\subset ℙ^n$$$ associated to the additive decompositions of degree $$$d$$$ forms in $$$n+1$$$ variables. Obviously $$$\rho(A)\le d$$$. We prove that $$$\rho(A)\ge d-k$$$ if $$$A$$$ spans $$$ℙ^n$$$ and $$$k$$$ is the maximal integer such that $$$x_0^k$$$ divides at least one monomial of $$$f$$$. If $$$f$$$ essentially depends on less variables, but $$$A$$$ spans $$$ℙ^n$$$, then $$$\rho(A)=d$$$. We give examples (but with $$$\#A$$$ bigger that the rank of $$$f$$$) in which we have $$$\rho(A)=d$$$.

Let $$$f$$$ be a degree $$$d$$$ form in $$$n+1$$$ variables $$$x_0,\dots ,x_n$$$. Any additive decomposition of $$$f$$$ is associated to a finite set $$$A\subset ℙ^n$$$ with $$$\#A$$$ the number of non-proportional addenda. We study the index of regularity $$$\rho(A)$$$ of $$$A$$$, i.e. the first integer $$$t$$$ such that $$$h^1(\mathcal{I}_A(t)) = 0$$$, of the finite subset $$$A\subset ℙ^n$$$ associated to the additive decompositions of degree $$$d$$$ forms in $$$n+1$$$ variables. Obviously $$$\rho(A)\le d$$$. We prove that $$$\rho(A)\ge d-k$$$ if $$$A$$$ spans $$$ℙ^n$$$ and $$$k$$$ is the maximal integer such that $$$x_0^k$$$ divides at least one monomial of $$$f$$$. If $$$f$$$ essentially depends on less variables, but $$$A$$$ spans $$$ℙ^n$$$, then $$$\rho(A)=d$$$. We give examples (but with $$$\#A$$$ bigger that the rank of $$$f$$$) in which we have $$$\rho(A)=d$$$.

index of regularity form additive decomposition

index of regularity form additive decomposition

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