Gröbner Bases of Generic Ideals

2017/11/14 by Capaverde, Juliane, Gao, Shuhong
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1711.05309

Abstract

Let I = ( f₁, …, f_n ) be a homogeneous ideal in the polynomial ring K[x₁, …,x_n] over a field K generated by generic polynomials. Using an incremental approach based on a method by Gao, Guan and Volny, and properties of the standard monomials of generic ideals, we show how a Gröbner basis for the ideal (f₁, …, f_i) can be obtained from that of (f₁, …, f_i-1). If deg f_i = d_i, we are able to give a complete description of the initial ideal of I in the case where d_i ≥ (∑_j=1^i-1d_j) - i -1. It was conjectured by Moreno-Socías that the initial ideal of I is almost reverse lexicographic, which implies a conjecture by Fröberg on Hilbert series of generic algebras. As a result, we obtain a partial answer to Moreno-Socías Conjecture: the initial ideal of I is almost reverse lexicographic if the degrees of generators satisfy the condition above. This result improves a result by Cho and Park. We hope this approach can be strengthened to prove the conjecture in full.

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