Main theorem of elimination theory: Difference between revisions

Revision as of 00:39, 16 May 2014

Revision as of 00:44, 16 May 2014

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Then:

:<math>\textstyle p(X) = \cap_d Z_d</math>.

Thus, it is enough to prove <math>Z_d</math> is closed. Let ''M'' be the matrix whose entries are coefficients of monomials of degree ''d'' in <math>x_i</math> in <math>x_0^{i_0} \cdots x_n^{i_n} f, \, i_0 + \dots i_n + \operatorname{deg}f</math> with homogeneous polynomials ''f'' in ''I''. Then the number of rowscolumns of ''M'' is the number of monomials of degree ''d'' in <math>x_i</math> (image a system of equations.) We allow ''M'' to have infinitely many rows.

Then <math>y \in Z_d \Leftrightarrow M(y)</math> has rank <math>< q \Leftrightarrow </math> all the <math>q \times q</math>-minors vanish at ''y''.