2022/06/19 by Sabine El Khoury, Khoury, Sabine El, Andrew R. Kustin +1
Computer Science · Mathematics · #13A02 #13C40 #13D02 #13E10 #13H10 #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2206.09473
openalex publication_date 2022/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let J be a quadratically presented grade three Gorenstein ideal in the standard graded polynomial ring R= k[x,y,z], where k is a field. Assume that R/J satisfies the weak Lefschetz property. We give the presentation matrix for J in terms of the coefficients of a Macaulay inverse system for J. (This presentation matrix is an alternating matrix and J is generated by the maximal order Pfaffians of the presentation matrix.) Our formulas are computer friendly; they involve only matrix multiplication; they do not involve multilinear algebra or complicated summations. As an application, we give the presentation matrix for J1=(xn+1,yn+1,zn+1):(x+y+z)n+1, when n is even and the characteristic of k is zero. Generators for J1 had been identified previously; but the presentation matrix for J1 had not previously been known. The first step in our proof is to give improved formulas for the presentation matrix of a linearly presented grade three Gorenstein ideal I in terms of the coefficients of the Macaulay inverse system for I.