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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

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Let f be a degree d form in n+1 variables x0,,xn. Any additive decomposition of f is associated to a finite set An with #A the number of non-proportional addenda. We study the index of regularity ρ(A) of A, i.e. the first integer t such that h1(IA(t))=0, of the finite subset An associated to the additive decompositions of degree d forms in n+1 variables. Obviously ρ(A)d. We prove that ρ(A)dk if A spans n and k is the maximal integer such that x0k divides at least one monomial of f. If f essentially depends on less variables, but A spans n, then ρ(A)=d. We give examples (but with #A bigger that the rank of f) in which we have ρ(A)=d.

Let f be a degree d form in n+1 variables x0,,xn. Any additive decomposition of f is associated to a finite set An with #A the number of non-proportional addenda. We study the index of regularity ρ(A) of A, i.e. the first integer t such that h1(IA(t))=0, of the finite subset An associated to the additive decompositions of degree d forms in n+1 variables. Obviously ρ(A)d. We prove that ρ(A)dk if A spans n and k is the maximal integer such that x0k divides at least one monomial of f. If f essentially depends on less variables, but A spans n, then ρ(A)=d. We give examples (but with #A bigger that the rank of f) in which we have ρ(A)=d.

index of regularity form additive decomposition

index of regularity form additive decomposition

encart iste group