Bound on the multiplicity of almost complete intersections

TL;DR

This paper establishes an upper bound on the multiplicity of almost complete intersection ideals in polynomial rings, relating it to the degrees of generators and the height of the ideal.

Contribution

It provides a new explicit upper bound for the multiplicity of almost complete intersections based on generator degrees and height.

Findings

Derived an explicit multiplicity bound for almost complete intersections.

The bound depends on degrees of generators and the height of the ideal.

Applicable in characteristic zero polynomial rings.

Abstract

Let $R$ be a polynomial ring over a field of characteristic zero and let $I \subset R$ be a graded ideal of height $N$ which is minimally generated by $N+1$ homogeneous polynomials. If $I=(f_1,...,f_{N+1})$ where $f_i$ has degree $d_i$ and $(f_1,...,f_N)$ has height $N$, then the multiplicity of $R/I$ is bounded above by $\prod_{i=1}^N d_i - \max\{1, \sum_{i=1}^N (d_i-1) - (d_{N+1}-1) \}$.