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Maximal minors of a matrix with linear form entries. (English) Zbl 1320.15014
Let \(P\) be a matrix whose entries are homogeneous polynomials in \(n\) variables of degree one over an algebraically closed field. The main theorem of this paper says that the maximal minors (say, \(m\)-minors) of \(P\) generate the linear space of homogeneous polynomials of degree \(m\) if \(P\) has the maximal rank \(m\) at every point of the affine \(n\)-space except for the origin. A counterexample shows that the result does not hold if the field is not algebraically closed.
15A54 Matrices over function rings in one or more variables
15A15 Determinants, permanents, traces, other special matrix functions
13D02 Syzygies, resolutions, complexes and commutative rings
Full Text: DOI
[1] DOI: 10.1007/978-3-642-10455-8 · Zbl 1246.35005 · doi:10.1007/978-3-642-10455-8
[2] Hartshorne R, Vol. 52 of graduate texts in mathematics (1977)
[3] Atiyah MF, Introduction to commutative algebra (1969)
[4] DOI: 10.1007/978-3-642-18808-4 · doi:10.1007/978-3-642-18808-4
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