The Hilbert function of some Hadamard products
type: Item
instance of: scholarly article
title: The Hilbert function of some Hadamard products
zbMATH Open document ID: 1400.14038
author: Xianqiang Yang
published in: Collectanea Mathematica
publication date: 2018-05-24T00:00:00Z
review text: The Hadamard product \(X*Y\)of two projective varieties \(X,Y\subset\mathbb P^n\) is the closure of the image of the rational map which sends \((x_0,\dots,x_n)\times (y_0,\dots,y_n)\in X\times Y\) to \((x_0y_0,\dots,x_ny_n)\). In geometric terms, it corresponds to a very special projection of the Segre product. Several applications of the Hadamard product to the statistics of Boltzmann machines and to matrix analysis suggest that a deeper knowledge of the geometric properties of \(X*Y\) can have relevant impact in applied algebraic geometry. In the paper under review, the authors study the Hilbert function (\(HF\)) of \(X*Y\). The main result says that \(HF(X*Y)\) is equal to the product \(HF(X)\cdot HF(Y)\) when either both \(X,Y\) are linear subvarieties, or when both \(X,Y\) have dimension at most \(1\).
reviewed by: Luca Chiantini
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 6872991
zbMATH Keywords: Hadamard product
MaRDI profile type: Publication
full work available at URL: s13348-017-0200-z
OpenAlex ID: W2626448909
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DOI: 10.1007/S13348-017-0200-Z
description: scientific article; zbMATH DE number 6872991