3
votes
1answer
96 views

Is $\mathbb R[x,y]_+$ countably generated as a quadratic module?

Let $\mathbb R[x,y]_+$ denote the set of positive polynomials in two variables. My problem can be stated as follows: Does there exist a countable set $M\subseteq \mathbb R[x,y]_+$ such that ...
6
votes
0answers
59 views

Bounding volume of cell in complement of zero set

I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...
0
votes
1answer
315 views

Degree of a real algebraic variety and regular morphisms

I'm reading Fulton's "Intersection theory", which i need for some applied needs. And i have two questions on general definition of degree used in Fulton. 1)Let us we have a real algebraic variety ...
2
votes
3answers
712 views

Isolated zero set of real polynomial in two variables

The polynomial $x^2+y^2$ has an isolated zero at the origin. And so do powers $(x^2+y^2)^n$ of this polynomial. I'm wondering if this is a special property of these real polynomials. Here's the ...
10
votes
3answers
1k views

Counting roots: multidimensional Sturm's theorem

Sturm's theorem gives a way to compute the number of roots of a one-variable polynomial in an interval [a,b]. Is there a generalization to boxes in higher dimensions? Namely, let $P_1,\dotsc,P_n\in ...
9
votes
3answers
429 views

Effective algorithm to test positivity

Let $f(x_1,\ldots, x_n)$ be a real polynomial in several variables. Is there an effective algorithm to test whether $f$ is positive (or nonnegative) on the whole of ${\mathbb{R}}^n$?