Mixed Toric Residues and Calabi-Yau Complete Intersections

Using Cayley trick, we define the notions of mixed toric residues and mixed Hessians associated with r Laurent polynomials \(f_1,\ldots,f_r\). We conjecture that the values of mixed toric residues on the mixed Hessians are determined by mixed volumes of the Newton polytopes of \(f_1,\ldots,f_r\). Using mixed toric residues, we generalize our Toric Residue Mirror Conjecture to the case of Calabi-Yau complete intersections in Gorenstein toric Fano varieties obtained from nef-partitions of reflexive polytopes.