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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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##### References:
 [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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