I have the conjecture stated below, which would be very helpful for a cryptographic security proof I'm working on for several months now. At first glance, it seemed a pretty natural question to me, but after asking a lot of people and also searching the internet I'm still without a solution or any helpful hints. This is the conjecture:


Let any multivariate polynomial $f\in\mathbb R[X\_1,\ldots,X\_n]$ be given, such that the corresponding variety $V\_f$ is not empty: $V\_f=\{x\in\mathbb R^n\,|\,f(x)=0\}\neq\emptyset$. Then there exists some constant $c>0$, such that for all $x\in\mathbb R^n$ one can estimate the distance of $x$ from $V$ as follows: $c\cdot\min\_{y\in V\_f}\|x-y\|^{\deg f}\leq|f(x)|$.


In particular, I only need this to hold true if $\deg(f)\leq4$ and $x$ is only taken from a compact set $K\subset\mathbb R^n$. If the conjecture turns out true, it can be used for generalization of a [classification result of mine](http://eprint.iacr.org/2010/654 "Completeness Theorems") in theoretic cryptography. Here are some further notes:


* Since all norms on $\mathbb R^n$ are equivalent, the assertion either holds true for every norm or for none.
* If $K\cap V\_f=\emptyset$, one can just set $c:=\min\_{x\in K,y\in V\_f}|f(x)|\cdot\|x-y\|^{-\deg f}$. So, the "interesting" values $x$ are close to $V$.
* If the conjecture is not true, then there exists a counterexample with irreducible polynomial $f$. Given any reducible counterexample, say $f=g\cdot h$, we can estimate $\min\_{y\in V\_f}\|x-y\|^{\deg f}$ by $\min\_{y\in V\_g}\|x-y\|^{\deg g}\cdot\min\_{y\in V\_h}\|x-y\|^{\deg h}$ and it follows straightforwardly that $g$ or $h$ must be a counterexample, too.
* If $n=1$, i.e. $f$ is univariate, the conjecture is true, since the only irreducible polynomials with non-empty variety are of the form $f(x)=ax+b$. Clearly, these are no counterexamples and thus, as mentioned above, there don't exist any univariate counterexamples at all.
* I was able to prove the weak variant of the conjecture with $x\in K$ for the case that $\deg(f)\leq2$, but the proof is quite long; it can be found [here](http://www.iks.kit.edu/fileadmin/User/Publikationen/Lemma_Distance_from_Quadric.pdf "Lemma").



Estimating the distance from a variety by evaluating the polynomial
You want this: <http://en.wikipedia.org/wiki/%C5%81ojasiewicz_inequality>


The exponent might not be the degree, though.


