TY  - Type of reference 
TI  - On the index of regularity of additive decompositions of forms 
AU  - Edoardo Ballico 
AB  - 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 $$$\rho(A)=d$$$. 
DO  - 10.21494/ISTE.OP.2025.1259 
JF  - Advances in Pure and Applied Mathematics 
KW  - index of regularity, form, additive decomposition, index of regularity, form, additive decomposition,  
L1  - https://openscience.fr/IMG/pdf/iste_apam25v16n2_1.pdf 
LA  - en 
PB  - ISTE OpenScience 
DA  - 2025/03/20 
SN  - 1869-6090 
TT  - Sur l’index de régularité des décompositions additives des formes 
UR  - https://openscience.fr/On-the-index-of-regularity-of-additive-decompositions-of-forms 
IS  - Issue 2 (March 2025) 
VL  - 16 
ER  -