Preprint
Powers of matrices with all principal minors equal to 1
ArXiv.org
27 Jun 2026
Abstract
Consider a square matrixAwhose all principal minors are equal to1 . Over a field, this property is inherited by any power ofA , but this is not the case over an arbitrary commutative ring. We show that it is the case over any regular ring, and also over the ringℤ / dfor any integerd , and in some other settings (quotients of Prüfer domains and principal quotients of normal domains). This generalizes Problem B5 of the 2021 Putnam contest. Over arbitrary commutative rings, we identify a stronger property that is always inherited by powers: We say that a matrixA = \left{(}{a}{_(i,j)}\right{)}{_(i,j∈\left{[}{n}{\right]})}{}is strongly1 -principled if all its diagonal entries are1and if all the cyclic productsa_(i₁, i₂) a_(i₂, i₃) ⋯ a_(i_(k), i₁)withk>1vanish. We show that the latter products are always integral over the ideal generated by the principal minors ofAminus1 .
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Details
- Title
- Powers of matrices with all principal minors equal to 1
- Creators
- Darij Grinberg
- Publication Details
- ArXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991022194943304721