On polynomials with given Hilbert function and applications

Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for the dimension of cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur.

Homing Plus programme of Foundation for Polish Science, co-financed from European Union, Regional Development Fund; Northeastern University; Project Galaad of INRIA Sophia Antipolis Méditerranée, France, Marie Curie Intra-European Fellowships for Carrer Development (FP7-PEOPLE-2009-IEF): ``DECONSTRUCT'', GNSAGA of INDAM, Mathematical Department Giuseppe Peano of Turin; Politecnico of Turin; Warsaw Center of Mathematics and Computer Science; Polish program KNOW; Polish National Science Center, project 2014/13/N/ST1/02640; ``Computational complexity, generalised Waring type problems and tensor decompositions'' project within ``Canaletto'', the executive program for scientific and technological cooperation between Italy and Poland, 2013-2015; Fundação para a Ciência e Tecnologia, projects ``Geometria Algébrica em Portugal'', PTDC/MAT/099275/2008, ``Comunidade Portuguesa de Geometria Algébrica'', PTDC/MAT-GEO/0675/2012, and sabbatical leave grant SFRH/BSAB/1392/2013; CIMA -- Centro de Investigação em Matemética e Aplicações, Universidade de Évora, projects PEst-OE/MAT/UI0117/2011 and PEst-OE/MAT/UI0117/2014; Fundação de Amparo à Pesquisa do Estado de São Paulo, grant 2014/12558--9; RCN project no 239015 ``Special Geometries''