Talk:Hadamard's lemma

Nullstellensatz
One direction to follow up on is possible ties to Hilbert's Nullstellensatz (weak version): a maximal ideal in the ring of polynomials in the variables $$ x_1, x_2, \dotsc, x_n$$ (over an algebraically closed field) has the form $$\langle x_1-a_1, x_2-a_2, \dotsc, x_n-a_n\rangle$$. This yields a conclusion similar to Haramard's lemma as follows. Let $$f$$ be a polynomial and $$a$$ an arbitrary point. Since $$f(x)-f(a)$$ has a zero at $$a$$, the ideal it generates cannot be the whole polynomial ring, and hence $$f(x)-f(a)$$ must be contained in some maximal ideal $$\langle x_1-a_1, x_2-a_2, \dotsc, x_n-a_n\rangle$$. This is equivalent to there being polynomials $$g_1, g_2, \dotsc, g_n$$ such that
 * $$ f(x) - f(a) = \sum_{i=1}^n (x_i-a_i) g_i(x)$$ identically for all $$x$$,

which is the same relation as in the Hadamard lemma. 130.239.235.159 (talk) 17:03, 5 November 2012 (UTC)