@ARTICLE{10.21494/ISTE.OP.2025.1259, 

TITLE={On the index of regularity of additive decompositions of forms}, 
  
AUTHOR={Edoardo Ballico, }, 
	
JOURNAL={Advances in Pure and Applied Mathematics}, 
	
VOLUME={16}, 

NUMBER={Issue 2 (March 2025)}, 
	
YEAR={2025}, 

URL={https://openscience.fr/On-the-index-of-regularity-of-additive-decompositions-of-forms}, 
	
DOI={10.21494/ISTE.OP.2025.1259}, 
	
ISSN={1869-6090}, 
	
ABSTRACT={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 $$$\rho(A)=d$$$.}}