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)