Main theorem of elimination theory: Difference between revisions

Revision as of 00:46, 11 March 2018

Revision as of 09:31, 11 March 2018

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Otherwise, let <math>d_i</math> be the degree of <math>f_i,</math> and suppose that the indices are chosen in order that <math>d_2\ge d_3 \ge\cdots\ge d_k\ge d_1.</math> The degree

:<math>D= d_1+d_2+\cdots+d_n-n+1 = 1+\sum_{i=1}^n (d_i-1)</math>

is called ''Macaulay's degree'' ofor ''Macaulay's bound'' because Macaulay's has proved that <math>\varphi(f_1),\ldots, \varphi(f_k)</math> have a non-trivial common zero if and only if the rank of the Macaulay matrix in degree {{mvar|D}} is lower than the number to its rows. In other words, the above {{mvar|d}} may be chosen once for all as equal to {{mvar|D}}.

Therefore, the ideal <math>\mathfrak r,</math> whose existence is asserted by the main theorem of elimination theory, is the zero ideal if {{math|''k'' < ''n''}}, and, otherwise, is generated by the maximal minors of the Macaulay matrix in degree {{mvar|D}}.