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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</math>
with homogeneous polynomials ''f'' in ''I'' and <math>i_0 + \dots i_n + \operatorname{deg}f = d</math>. Then the number of columns of ''M'', denoted by ''q'', 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''.
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