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On partially matchable subspaces in a field extension

  • Mohsen Aliabadi
  • Jozsef Losonczy
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Abstract

We develop a theory of partial matchings between finite-dimensional subspaces A,B in a field extension K subsetneq L, providing structural, quantitative, and extremal results for their deficiency. Our main results include (1) a characterization of those pairs (A,B) that are partially matchable up to a specified defect, (2) a decomposition theorem for pairs (A,B) having positive deficiency, (3) an existence criterion for pairs having a prescribed dimension and satisfying a deficiency bound, and (4) a formula for the maximum attainable deficiency at each dimension, assuming 1 ∉ B and the extension has a proper nontrivial intermediate field of finite K-dimension. We use these results to recover and extend various parts of this area of matching theory. Our approach blends algebraic techniques with tools from matroidal transversal theory, and utilizes a linearized version of the e-transform from additive number theory.

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Cite

BibTeXRIS
BibTeX
@misc{aliabadi2026partially,
  title = {{On partially matchable subspaces in a field extension}},
  author = {Mohsen Aliabadi and Jozsef Losonczy},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2606.12725},
  eprint = {2606.12725},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.12725},
}
RIS
TY  - UNPB
TI  - On partially matchable subspaces in a field extension
AU  - Mohsen Aliabadi
AU  - Jozsef Losonczy
PY  - 2026
DA  - 2026/08/27
PB  - arXiv
DO  - 10.48550/arxiv.2606.12725
UR  - https://arxiv.org/abs/2606.12725
AB  - We develop a theory of partial matchings between finite-dimensional subspaces A,B in a field extension K subsetneq L, providing structural, quantitative, and extremal results for their deficiency. Our main results include (1) a characterization of those pairs (A,B) that are partially matchable up to a specified defect, (2) a decomposition theorem for pairs (A,B) having positive deficiency, (3) an existence criterion for pairs having a prescribed dimension and satisfying a deficiency bound, and (4) a formula for the maximum attainable deficiency at each dimension, assuming 1 ∉ B and the extension has a proper nontrivial intermediate field of finite K-dimension. We use these results to recover and extend various parts of this area of matching theory. Our approach blends algebraic techniques with tools from matroidal transversal theory, and utilizes a linearized version of the e-transform from additive number theory.
ER  -