Mertens theorem (resultant system)

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The set of Mertens is a set of homogeneous polynomials over, inter alia, in the algebraic geometry for projective algebraic quantities is relevant.

formulation

Let K be an algebraically closed field and be homogeneous polynomials of degrees :

Then there is a resultant system , i.e. polynomials in the indeterminate , so that the polynomials have a common zero other than 0 (i.e. one in projective space) if and only if for all k

proof

have no common zero except 0 if and only if their common zero set is contained in the zero set of, which is only 0. As one can easily consider with the help of Hilbert's zero theorem , this is the case if and only if there is a natural number d such that for all multi-indices with magnitude d. So all monomials of degree d are contained in this ideal, so they can be represented as the sum of these polynomials, which are homogeneous with other without restriction, as coefficients. Among the , whereby there must be as many linearly independent as monomials of degree d. So the following applies: They have no common zero except for 0 if and only if for all natural numbers d each (number of monomials of degree d) the are linearly independent. This is equivalent to the disappearance of sub-determinants formed from 0 and . These are polynomials in and because of the noetherzeity of ( Hilbert's basis theorem ), finitely many are sufficient.

literature

  • Franz Mertens : On the theory of elimination, session report. Vienna 108 (1889), page 1174
  • BL van der Waerden : On the construction of the resultant system for homogeneous equations, in: Archiv der Mathematik, Volume 5, Numbers 4-6, pp. 371ff, 1954