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On polynomials with given Hilbert function and applications

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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.

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JJ and KR thank Jarosław Buczyński for fruitful discussions and the Homing Plus programme of Foundation for Polish Science, co-financed from European Union, Regional Development Fund for partial support of their mutual visits. PMM thanks Anthony Iarrobino and Jerzy Weyman for the invitation and hospitality at Northeastern University, and is grateful to Anthony Iarrobino for having introduced him to this subject and for very fruitful discussions. He also thanks James Adler for help with language. AB was partially supported by 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, Italy, and Politecnico of Turin, Italy. JJ is a doctoral fellow at the Warsaw Center of Mathematics and Computer Science financed by the Polish program KNOW and by Polish National Science Center, Project 2014/13/N/ST1/02640 and a member of “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. PMM was partially supported by 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, by CIMA—Centro de Investigação em Matemática e Aplicações, Universidade de Évora, Projects PEst-OE/MAT/UI0117/2011 and PEst-OE/MAT/UI0117/2014, and by Fundação de Amparo à Pesquisa do Estado de São Paulo, Grant 2014/12558–9. KR was supported by the RCN Project No 239015 “Special Geometries”.

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Correspondence to Pedro Macias Marques.

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Bernardi, A., Jelisiejew, J., Marques, P.M. et al. On polynomials with given Hilbert function and applications. Collect. Math. 69, 39–64 (2018).

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