Linear restrictions on cone polynomials

TL;DR

This paper investigates the limitations of cone polynomials in spanning the space of degree d polynomials vanishing on a point set in projective space, revealing specific restrictions based on degree and dimension.

Contribution

It establishes new conditions under which cone polynomials do not fully span the polynomial space, highlighting restrictions for odd degrees and certain dimensions.

Findings

Cone polynomials do not span the entire polynomial space for odd degrees d ≥ 3.

They span a subspace of codimension at least two when n=2, d ≡ 1 mod 4, and d ≥ 5.

The results specify the limitations of cone polynomials in polynomial interpolation problems.

Abstract

For a set $S$ of $d$ points in the $n$-dimensional projective space over a field of characteristic zero, we prove that the polynomials of degree $d$ whose zero sets are cones over $S$ do not span the vector space of polynomials of degree $d$ vanishing on $S$, if $d$ is odd and $d\ge 3$. Furthermore, they span a subspace of codimension at least two, if $n=2$, $d=1\pmod 4$ and $d\ge 5$.