On the index of regularity of additive decompositions of forms

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