Some Questions on Algebraic Varieties

Question

I'm doing some research on algebraic varieties for an engineering application. I am totally new to the subject and have a general (B.S.-level) background in math. I am currently using Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea as my main reference.

I am interested in two basic questions that don't seem to be covered or easily found in that book, and hope that a colleague can point me to literature in which these questions are discussed more completely.

1. My most basic question: Let $\{f_1(x_1,\dots,x_p),\cdots,f_m(x_1,\dots,x_p)\}$ be a set of $m$ real valued polynomials in $p$ unknowns. Under what conditions does the system

\begin{align} % f_1(x_1,\dots,x_p) &= 0 \\ % &\vdots \\ % f_m(x_1,\dots,x_p) &= 0 \\ % \end{align}

have a unique solution? What about uniqueness over the real numbers only? There are a number of theorems that address how to reduce $m$ equations to $n \leq m$ independent equations via elimination theory, but this doesn't give much information about uniqueness, as the following example shows.

Example. Consider the system $x_1^2 + x_2^2 = 0$. The unique solution is $(x_1,x_2) = 0$. Next, consider the system $x_1^2 + x_2^2 - 1 = 0$. Any point on the unit circle is a solution to this system, hence there is no unique solution. Note however that both systems consist of exactly one independent equation.

I also thought that dimension theory might help me answer this question, but I'm not sure simply determining the dimension of the variety determined by $f_1,\dots,f_m$ is enough to deduce the uniqueness of the solution. For example, the dimension of the variety determined by the quadratic $(x-2)(x-3)$ is, I believe, zero, but the solution is still not unique.

2. Are there general results regarding the Lebesgue measure of an affine variety? For example, the plane curves in $\mathbb{R}^2$ all have Lebesgue measure zero. What about higher dimensional varieties?

Answer 1

Question 1 can be a bit tricky. In the first case, where you allow complex solutions, the variety has to be of dimension zero and have exactly one irreducible component. One way to test this is to check if the ideal generated by $f_1,\cdots,f_m$ has a maximal ideal as it's radical. In particular, this means that $m\geq p$. One can test this using computer algorithms: given an ideal $(f_1,\cdots,f_m)$, one can ask for it's primary decomposition, and you want to get a unique maximal ideal out in order to have a unique point over the complex numbers.

Uniqueness over the reals is more difficult. The best general tool for investigating uniqueness over the reals is the cylindrical algebraic decomposition algorithm. It has fairly high computational complexity (doubly-exponential in the number of variables), but it is implementable on a computer, and it will give you a list of all real connected components, their local dimension, and a point on each of them. This will let you decide whether you in fact have a unique solution or not. Figuring out how to improve this algorithm is an active research area, so quick solutions outside of special cases are unlikely.

To speak to your mention of dimension theory, dimension theory won't be very helpful over the reals. The only connection between the dimension of the variety and the dimension of the real points is that the dimension of the real points can't be greater than the dimension of the variety. Standard algebro-geometric tools aren't really built for this situation- tools from semi-algebraic and o-minimal geometry like the cylindrical decomposition algorithm above are really where one should be looking to solve problems like this.

Question 2: For this question, we'll call a variety a subset of $\Bbb R^n$ given by the vanishing of a finite number of nonconstant polynomial functions. Any such variety has measure zero. One can see this as follows: if a real-analytic function defined on a connected set $U\subset \Bbb R^n$ is zero on a subset of positive measure, then it's identically zero.

What this means for us is that since any polynomial is analytic, nonconstant polynomials must have zero sets with measure zero.