Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 3 min ago
PreprintCC BY 4.0arXiv

Dimensions of permanental varieties in arbitrary size

  • Ying Xie
Open at sourcePDFDOICite

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.

Versions

Related

  1. The square function estimates for cones in arbitrary dimensions and elliptic quadrics
    Faruk Temur
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  2. Multivariable Painleve'-II equation: connection formulas for arbitrary system size
    Nikolai A. Sinitsyn
    arXivPreprintCC0Full textMathematical physicsCondensed matterarXiv
  3. Piecewise P2 finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation
    Rui Ma, Yang Xing, Shuo Zhang
    arXivPreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv
  4. On the size of the fibers of a morphism of algebraic varieties
    Abdelkader Mokkadem
    HALPreprintCC BY-NC 4.0Full textAlgebra & number theoryHALarXiv
  5. Regularity and Rivière's GL(m)-Gauge Construction for Elliptic Systems with Antisymmetric Potentials in Arbitrary Dimensions
    Carolin Bayer
    arXivPreprintarXiv licenceFull textAnalysis & PDEsGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{xie2026dimensions,
  title = {{Dimensions of permanental varieties in arbitrary size}},
  author = {Ying Xie},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27320},
  eprint = {2609.27320},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27320},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Dimensions of permanental varieties in arbitrary size
AU  - Ying Xie
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.27320
UR  - https://arxiv.org/abs/2609.27320
AB  - Let I_h(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]/I_h(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.
ER  -