Tropicalization and irreducibility of generalized Vandermonde determinants. (English) Zbl 1229.12005
The authors characterize the \(N\times n\) matrices \((\gamma_{jk})\) of nonnegative integers such that the generalized \(N\times N\) Vandermonde determinant \(\det (X_{i1}^{\gamma_{j1}}\cdots X_{in}^{\gamma_{jn}} )\) is irreducible over a given algebraically closed field. They also characterize the matrices \((\gamma_{jk})\) such that the tropicalization of the generalized Vandermonde determinant with respect to the variables of the first row and with respect to a suitable valuation is an irreducible tropical variety.
Reviewer: P. E. Frenkel (Budapest)
MSC:
| 12E10 | Special polynomials in general fields |
| 12E05 | Polynomials in general fields (irreducibility, etc.) |
| 14M12 | Determinantal varieties |
| 14T05 | Tropical geometry (MSC2010) |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
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