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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.
MSC:
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
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[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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