TITLE: product of nonsingular polynomials
QUESTION [1 upvotes]: In $R^n$, the polynomial $P$ is called nonsingular if $0$ is its regular value i.e. $\forall x  s.t. p(x)=0,\mbox{then} \nabla  p(x)\neq0$.
How to prove that the product of nonsingular polynomials has the property that its nonsingular points are dense in its zero set?
I have seen this property on Guth,s paper of polynomial partition. But I think it is wrong. If $$p1=p2=x,p=p1\cdot p2=x^2$$
It only has one singular point $0$ in zero set.
So, something must be wrong.

REPLY [0 votes]: The question is wrong usually. But if we ask the polynomials are prime to each other, the question is true. And in the paper of Guth, we can ask this property by the prime factorization of the product.Then delete the power.
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