Re: Measure of the roots of a multivariate polynomial.
Originally Posted by RaisinBread
Hey, I'm trying to show the following proposition, but I have no idea where to start with it;
Let be a polynomial, where . Show that the set has Lebesgue measure zero.
This only works for nonzero polynomials, where we can use induction. Suppose the result holds for and let . Then has measure zero in . Furthermore, for each fixed , the polynomial is nonzero, which means the set has measure zero in by the inductive hypothesis. Show that
both have measure zero in , and it will follow that has measure zero, where is the set of zeros of .