Suppose I have a collection of polynomials with multiple variables (more polynomials than variables, say), and I'm given noisy versions the values of these polynomials at a certain unknown point. I would like to solve for this point in a stable manner. What is known about this problem?

Certainly, the notion of stability should depend on the degree of the polynomials. I am particularly interested in case where the degree is 2 and the field is real, but I'm open to results involving higher degrees and algebraically closed fields.

**UPDATE:** Following Misha's suggestion, I'll describe one way to estimate the desired point. Take $m>n$ and let $F\colon\mathbb{R}^n\rightarrow\mathbb{R}^m$ be a polynomial mapping. I am given $y=F(x)+e$ for some noise vector $e\in\mathbb{R}^m$ of small norm, and my task is to estimate $x$. Here, we might find the closest point $\hat{y}$ in the image $F(\mathbb{R}^n)$ to $y$, and then solve the system $F(\hat{x})=\hat{y}$ for $\hat{x}$. But what can be said about the quality of this estimator? For example, how is $\|\hat{x}-x\|$ related to $\|e\|$? Is there a Holder-type relationship in terms of the degree of $F$?