Year: 2026
Type: preprint
Abstract: : Let Ih(a, b) be the ideal generated by the h × h permanents of a generic a × b matrix over a field of characteristic different from two. We prove that dim K[X]/Ih(a, b) = max(a, b)(h − 1) whenever 1 ≤ h ≤ min(a, b) and h < max(a, b). Consequently, the critical locus of the permanent of order n has codimension 2n for every n ≥ 2. We also prove that the maximal permanents of a generic k × (k + 1) matrix generate a geometrically reduced complete intersection. The dimension argument uses a pivot expansion and cubic relations in the border variables. Two successive comparisons with highest-degree equations reduce the relevant fibers to a coordinate-support calculation. Reducedness follows from a separate regular-sequence argument and a dimension estimate excluding components over the exceptional locus.
Source: arXiv (Cornell University)
Author: Xie, Ying
Institution: Kennesaw State University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214031399