Is the pushforward of Lebesgue measure on a cube by a polynomial piecewise smooth?

Question

Question: Let $Q \subset {\bf R}^n$ be a cube, and let $P: {\bf R}^n \to {\bf R}$ be a non-constant polynomial. Is it true that the pushforward $P_*( m\downharpoonright_{Q} )$ of Lebesgue measure on $Q$ by $P$ is (a) absolutely continuous, and (b) having a density that is piecewise smooth? Furthermore (c) does the density and all its derivatives blow up at an at most polynomial rate as one approaches the singularities?

Example 1: If $P: {\bf R}^2 \to {\bf R}$ is the multiplication map $P(x,y) = xy$, then the pushforward of Lebesgue measure on the square $[-1,1]^2$ is absolutely continuous and has a density of $t \mapsto 2\log \frac{1}{|t|} 1_{[-1,1]}(t)$ on this interval, which is piecewise smooth with a singularities at $t=-1,0,+1$, with all derivatives blowing up at most polynomially as one approaches any of these singularities.

Example 2: If $P: {\bf R}^2 \to {\bf R}$ is the addition map $P(x,y) = x+y$, then the pushforward of Lebesgue measure on the square $[0,1]^2$ is absolutely continuous and has a density of $t \mapsto (1-|1-t|) 1_{[0,2]}(t)$ on this interval, which is piecewise smooth with a singularities at $t=0,1,2$, with all derivatives blowing up at most polynomially (actually they are all bounded in this case) as one approaches any of these singularities. Note here that there are no critical points of $P$; the singularities are coming instead from the corners of the square.

This looks like a question that should be a quick consequence of some utterly standard theory, yet to my embarrassment I cannot even establish absolute continuity (though for my application it is the piecewise smoothness and polynomial blowup which is relevant). [EDIT: as noted in comments, Ehresmann's lemma (plus Sard's theorem) will suffice to get the absolute continuity at least.] A few things I tried (but failed) to apply:

• Sard's theorem and the Łojasiewicz inequality seem relevant, but insufficient by themselves to even establish absolute continuity (though the Łojasiewicz inequality does at least seem to provide some non-trivial regularity in a negative Sobolev space).
• Perhaps there is some sort of preparation theorem that provides a tractable analytic description of how the level sets of $P$ sit inside the cube $Q$? Resolution of singularities seems potentially useful in this regard (and is used for instance in some proofs of the Łojasiewicz inequality), but I don't see how else to apply it here.
• These pushforward measures look vaguely like some geometric analogue of a period, particularly if one generalizes from cubes to other polytopes or semi-algebraic sets, but I don't know where to take that observation further (especially since much of the theory of periods appears to be conjectural).
• In all the examples I can compute, the measure is not only piecewise smooth, but in fact piecewise analytic, which perhaps suggests some complexification of the problem, but I was not able to find such a complexification.
• It is tempting to try to first work out the moments $\int_Q P^m$, possibly suggesting that analytic combinatorics methods may be useful? One could also try looking at asymptotics for the oscillatory integrals $\int_Q e^{i\lambda P}$, although this seems to me to be replacing the problem with a strictly harder one.
• The problem also vaguely reminds me of the Tarski-Seidenberg theorem, but I don't see an analogue of quantifier elimination (as Example 1 already shows, eliminating a dimension tends to introduce transcendental functions to the situation).

Answer 1

$\newcommand{\la}{\lambda}\newcommand{\R}{\mathbb{R}}$Let $I:=(0,1)$ and $Q=I^n$. Let $\la$ denote the Lebesgue measure over $\R^n$. Let $P_j$ denote the partial derivative of $P$ wrt to its $j$th argument. Because the polynomial $P=P(x_1,\dots,x_n)$ is non-constant, at least one of the $P_j$'s is a nonzero polynomial. Without loss of generality, $P_1$ is a nonzero polynomial.

It follows by the main theorem of algebra and the Tonelli theorem that the Lebesgue measure $\la(N)$ of the (closed) set \begin{equation*} N:=\{(x_1,\dots,x_n)\in \R^n\colon P_1(x_1,\dots,x_n)=0\} \tag{10}\label{10} \end{equation*} of the zeros of the nonzero polynomial $P_1$ is $0$.

Consider now the transformation $$\R^n\ni(x_1,\dots,x_n)\mapsto g(x_1,\dots,x_n):=(P(x_1,\dots,x_n),x_2,\dots,x_n)\in\R^n, $$ the polynomials \begin{equation*} p_{(y_1,\dots,y_n)}(x_1):=P(x_1,y_2,\dots,y_n)-y_1, \end{equation*} the sets \begin{equation*} Z_{(y_1,\dots,y_n)}:=\{x_1\in \R\colon(x_1,y_2,\dots,y_n)\notin N, p_{(y_1,\dots,y_n)}(x_1)=0\}, \end{equation*} for $(y_1,\dots,y_n)\in\R^n$, the set \begin{equation*} Y:=\{(y_1,\dots,y_n)\in\R^n\colon p_{(y_1,\dots,y_n)} \text{ has no multiple roots}\}, \end{equation*} and the sets
\begin{equation*} A_m:=\{(y_1,\dots,y_n)\in Y\colon\text{card}\,Z_{(y_1,\dots,y_n)}=m\} \end{equation*} for $m=0,\dots,d$, where $\text{card}$ denotes the cardinality and $d$ is the degree of $P$ wrt $x_1$. Note that $(A_0,\dots,A_d)$ is a partition of $Y$, and the sets $A_1,\dots,A_m$ are open. By the main theorem of algebra and the Tonelli theorem, the open set $Y$ is of full Lebesgue measure.

By the implicit function theorem, for each $m\in\{0,\dots,d\}$, each $(y_1,\dots,y_n)\in A_m$, and each integer $k\in[1,m]$, there is a real-analytic function $X_{m,k}$ defined on a neighborhood $U_{(y_1,\dots,y_n)}$ of the point $(y_1,\dots,y_n)\in A_m$ such that \begin{equation*} g(X_{m,k}(z_1,\dots,z_n),z_2,\dots,z_n)=(z_1,\dots,z_n) \end{equation*} for all $(z_1,\dots,z_n)\in U_{(y_1,\dots,y_n)}$, and the value of the Jacobian determinant of the map \begin{equation*} U_{(y_1,\dots,y_n)}\ni(z_1,\dots,z_n)\mapsto(X_{m,k}(z_1,\dots,z_n),z_2,\dots,z_n) \end{equation*} at the point $(z_1,\dots,z_n)$ is \begin{equation*} \partial_{z_1}X_{m,k}(z_1,\dots,z_n)=\frac1{P_1(X_{m,k}(z_1,\dots,z_n),z_2,\dots,z_n)}. \end{equation*}

So, by the change of variables in multiple integrals, the density (say $f$) of the pushforward $g_\#(\la\downharpoonright Q)$ under $g$ of the measure $\la\downharpoonright Q$ wrt to $\la$ is given by the formula \begin{equation*} f(y_1,\dots,y_n)=\sum_{m=1}^d \sum_{k=1}^m\frac{1((y_1,\dots,y_n)\in B_{m,k})}{|P_1(X_{m,k}(y_1,\dots,y_n),y_2,\dots,y_n)|} \tag{20}\label{20} \end{equation*} for $(y_1,\dots,y_n)\in\R^n\setminus g(N)$, where \begin{equation*} B_{m,k}:=\{(y_1,\dots,y_n)\in A_m\colon(X_{m,k}(y_1,\dots,y_n),y_2,\dots,y_n)\in Q\}. \end{equation*} Note that $\la(g(N))=0$, since $\la(N)=0$ and $g$ is Lipschitz. So, we can let $f:=0$ on $g(N)$.

Now, the density (say $f_1$) of the pushforward $P_\#(\la\downharpoonright Q)$ of the measure $\la\downharpoonright Q$ under $P$ wrt to $\la$ is given by the formula \begin{equation*} f_1(y_1)=\int_{\R^{n-1}}dy_2\,\cdots\, dy_n\,f(y_1,\dots,y_n) \tag{30}\label{30} \end{equation*} for $y_1\in\R$.

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More detailed information can be obtained using cylindrical algebraic decomposition and the Tarski–Seidenberg theorem.

Indeed, using cylindrical algebraic decomposition, one can partition $\R^n$ into finitely many connected semialgebraic sets $(C_i)_{i\in I}$ called cells, on which the polynomial $P_1$ has constant sign: $+$, $-$, or $0$, such that for any $i$ and $j$ in $I$ one has either $\pi(C_i)=\pi(C_j)$ or $\pi(C_i)\cap\pi(C_j)=\emptyset$, where $\pi$ is the projection of $\R^n$ onto $\R$ consisting in removing the last $n-1$ coordinates.

Moreover, by the Tarski–Seidenberg theorem, $\pi(C_i)$ is a semialgebraic set for each $i\in I$.

Thus, we have a partition $(I_k)_{k\in K}$ of the finite set $I$ and a partition $(D_k)_{k\in K}$ of $\R$ into semialgebraic sets $D_k$ such that for each $k\in K$ \begin{equation*} D_k=\pi(C_i)\text{ for }i\in I_k. \end{equation*}

For each $k\in K$, the semialgebraic subset $(0,1)\cap D_k$ of $\R$ is the finite disjoint union of intervals $D_{k,l}$: \begin{equation*} (0,1)\cap D_k=\;\cdot \hspace{-10pt}\bigcup_{l=1}^{L_k} D_{k,l}. \end{equation*} Introducing now the semialgebraic sets \begin{equation*} C_{k,i,l}:=C_i\cap\pi^{-1}(D_{k,l})\text{ for }i\in I_k \end{equation*} $k\in K$, and $l\in[L_k]$, we get the finite partition $(C_{k,i,l}\colon k\in K, i\in I_k, l\in[L_k])$ of $(0,1)\times\R^{n-1}$ into connected semialgebraic sets and the finite partition $(D_{k,l}\colon k\in K, l\in[L_k])$ of the interval $(0,1)$ into intervals $D_{k,l}$ such that \begin{equation*} D_{k,l}=\pi(C_{k,i,l}) \end{equation*} for all $k\in K, i\in I_k, l\in[L_k]$. In each of the $C_{k,i,l}$'s, the sign (say $s_i$) is constant. Also, the union of all $C_{k,i,l}$'s with $s_i=0$ is contained in the set $N$ defined by \eqref{10}, and $\la(N)=0$, as was established previously.

So (cf. \eqref{20}), \begin{equation*} f(y_1,\dots,y_n)=\sum_{k=1}^K \sum_{i\in I_k}\sum_{l=1}^{L_k} \frac{1((y_1,\dots,y_n)\in C_{k,i,l})}{|P_1(X_{k,i,l}(y_1,\dots,y_n),y_2,\dots,y_n)|} \tag{40}\label{40} \end{equation*} for certain real-analytic functions $X_{k,i,l}$ defined on the corresponding cells $C_{k,i,l}$ $(y_1,\dots,y_n)\in(0,1)\times\R^{n-1}\setminus g(N)$. Again, we can let $f:=0$ on $(0,1)\times\R^{n-1}\cap g(N)$.
It follows from \eqref{40} that $f$ can explode near the boundaries of cells $C_{k,i,l}$ only polynomially. So, by \eqref{30}, $f_1$ can explode near the endpoints of the intervals $D_{k,l}$ only polynomially.