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