Edit: One can also assume $M,N$ compact in the following, and that $M$ is equipped with a Riemannian metric as well.

Sard's theorem says that for every smooth map $f: M \to N$ between two manifolds, the set of critical values of $f$ is a set of measure $0$ in $N$. Now assume $N$ is equipped with a Riemannian metric, and let $A$ be the set of critical values of $f$, and $T_\epsilon(A)$ the tubular neighborhood of $A$ in $N$ with radius $\epsilon$, are there any results on the volume growth of $T_\epsilon(A)$ under certain conditions on $f$? For instance, $f$ could be an algebraic map of bounded degree (in the sense of the lowest degree of a representing polynomial of $f$ under suitable coordinates), which is in fact the case I care about.

If $f$ is an algebraic map between two algebraic varieties $M,N$ that are also smooth manifolds, is it true that the set of critical values of $f$ is a union of submanifolds of $N$ with dimension strictly less than $\dim N$? If that's the case, then it appears the volume growth can be estimated once we know the Riemannian induced volume of each of those lower dimensional component. But maybe there is a way to estimate volume growth more intrinsically?

Edit: One can also assume $M,N$ compact in the following, and that $M$ is equipped with a Riemannian metric as well. Incorporated Mike's suggestion.

Sard's theorem says that for every smooth map $f: M \to N$ between two manifolds, the set of critical values of $f$ is a set of measure $0$ in $N$. Now assume $N$ is equipped with a Riemannian metric, and let $A$ be the set of critical values of $f$, and $T_\epsilon(A)$ the tubular neighborhood of $A$ in $N$ with radius $\epsilon$, are there any results on the volume growth of $T_\epsilon(A)$ under certain conditions on $f$? For instance, $f$ could be an algebraic map of bounded degree (in the sense of the lowest degree of a representing polynomial of $f$ under suitable coordinates), which is in fact the case I care about.

If $f$ is an algebraic map between two algebraic varieties $M,N$ that are also smooth manifolds, is it true that the set of critical values of $f$ is a union of submanifolds (or perhaps closed submanifolds with boundary) of $N$ with dimension strictly less than $\dim N$? If that's the case, then it appears the volume growth can be estimated once we know the Riemannian induced volume of each of those lower dimensional component. But maybe there is a way to estimate volume growth more intrinsically?

Sard's theorem says that for every smooth map $f: M \to N$ between two manifolds, the set of critical values of $f$ is a set of measure $0$ in $N$. Now assume $N$ is equipped with a Riemannian metric, and let $A$ be the set of critical values of $f$, and $T_\epsilon(A)$ the tubular neighborhood of $A$ in $N$ with radius $\epsilon$, are there any results on the volume growth of $T_\epsilon(A)$ under certain conditions on $f$? For instance, $f$ could be an algebraic map of bounded degree, which is in fact the case I care about.

If $f$ is an algebraic map between two algebraic varieties $M,N$ that are also smooth manifolds, is it true that the set of critical values of $f$ is a union of submanifolds of $N$ with dimension strictly less than $\dim N$? If that's the case, then it appears the volume growth can be estimated once we know the Riemannian induced volume of each of those lower dimensional component. But maybe there is a way to estimate volume growth more intrinsically?

Edit: One can also assume $M,N$ compact in the following, and that $M$ is equipped with a Riemannian metric as well.

Sard's theorem says that for every smooth map $f: M \to N$ between two manifolds, the set of critical values of $f$ is a set of measure $0$ in $N$. Now assume $N$ is equipped with a Riemannian metric, and let $A$ be the set of critical values of $f$, and $T_\epsilon(A)$ the tubular neighborhood of $A$ in $N$ with radius $\epsilon$, are there any results on the volume growth of $T_\epsilon(A)$ under certain conditions on $f$? For instance, $f$ could be an algebraic map of bounded degree (in the sense of the lowest degree of a representing polynomial of $f$ under suitable coordinates), which is in fact the case I care about.

If $f$ is an algebraic map between two algebraic varieties $M,N$ that are also smooth manifolds, is it true that the set of critical values of $f$ is a union of submanifolds of $N$ with dimension strictly less than $\dim N$? If that's the case, then it appears the volume growth can be estimated once we know the Riemannian induced volume of each of those lower dimensional component. But maybe there is a way to estimate volume growth more intrinsically?