Multigraded Sylvester forms, Duality and Elimination Matrices

Laurent Busé (1) , Marc Chardin (2) , Navid Nemati (1)

1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

In this paper we study the equations of the elimination ideal associated with n+1 generic multihomogeneous polynomials defined over a product of projective spaces of dimension n. We first prove a duality property and then make this duality explicit by introducing multigraded Sylvester forms. These results provide a partial generalization of similar properties that are known in the setting of homogeneous polynomial systems defined over a single projective space. As an important consequence, we derive a new family of elimination matrices that can be used for solving zero-dimensional multiprojective polynomial systems by means of linear algebra methods.

Domaines: Algèbre commutative [math.AC]; Calcul formel [cs.SC]; Géométrie algébrique [math.AG]

Laurent Busé, Marc Chardin, Navid Nemati. Multigraded Sylvester forms, Duality and Elimination Matrices. Journal of Algebra, 2022, 609 (1), pp.514-546. ⟨10.1016/j.jalgebra.2022.06.022⟩. ⟨hal-03202525v2⟩