Semialgebraic Solution
of Linear Equation with
Continuous Semialgebraic Coefficients

Marcello Malagutti

1 Introduction

C. Fefferman gave in [1], by means of analysis techniques, a necessary and sufficient condition for the existence of a continuous solution $(\phi_{1},\cdots,\phi_{s})$ of the system

\phi=\sum\limits_{i=1}^{s}\phi_{i}f_{i}

(1.1)

given the continuous functions $\phi$ and $f_{i}$ . More precisely, C. Fefferman, applying the theory of the Glaeser refinements for bundles, proved that system (1.1) has a continuous solution iff the affine Glaeser-stable bundle associated with system (1.1) has no empty fiber.

Moreover J. Kollár, in the same (joint) paper [1], starting from the above result and making use of algebraic geometry techniques as blowing up and at singular points, proved that fixed the polynomials $f_{1},...,f_{s}$ and assuming system (1.1) solvable, then:

1) if $\phi$ is semialgebraic then there is a solution $(\psi_{1},\cdots,\psi_{s})$ of $\phi=\underset{i}{\sum}\psi_{i}f_{i}$ such that the $\psi_{i}$ are also semialgebraic;

2) let $U\subset\mathbb{R}^{n}\backslash Z$ (where $Z:=(f_{1}=\cdots=f_{r}=0)$ ) be an open set such that $\phi$ is $C^{m}$ on $U$ for some $1\leq m\leq\infty$ or $m=\omega$ . Then there is a solution $\psi=(\psi_{1},\cdots,\psi_{s})$ of $\phi=\sum\limits_{i=1}^{s}\psi_{i}f_{i}$ such that the $\psi_{i}$ are also $C^{m}$ on $U$ .

In this paper we generalise and prove, by using Fefferman’s techniques, a part of the above results shown by Kollár. More in details, we consider a compact metric space $Q\subseteq\mathbb{R}^{n}$ and a system of linear equations

A\left(x\right)\phi\left(x\right)=\gamma\left(x\right),\quad x=(x_{1},\ldots,x_{n})\in Q

(1.2)

where

Q\ni x\longmapsto A(x)=(a_{ij}(x))\in M_{r,s}(\mathbb{R})

is continuous semialgebraic, with $M_{r,s}(\mathbb{R})$ denoting the set of real $r\times s$ matrices and

Q\ni x\longmapsto\gamma(x)\in\mathbb{R}^{r},\gamma(x)=\left[\begin{array}[]{c}\gamma_{1}(x)\\ \vdots\\ \gamma_{r}(x)\end{array}\right]\in\mathbb{R}^{r}

being themselves continuous semialgebraic functions on $Q\subseteq\mathbb{R}^{n}$ .

Our aim is to find a necessary and sufficient condition for the existence of a solution $Q\ni x\longmapsto\phi\left(x\right)=\left[\begin{array}[]{c}\phi_{1}(x)\\ \vdots\\ \phi_{s}(x)\end{array}\right]\in\mathbb{R}^{s}$ of system (1.2), with the $\phi_{i}:Q\rightarrow\mathbb{R}$ continuous and semialgebraic.

In particular, we find that a continuous and semialgebraic solution exists if and only if there exits a continuous solution and a semialgebraic one (they may possibly be different) under the hypothesis that

B(x,r_{v_{x}})\ni y\longmapsto\gamma_{v_{x}}(y)=\varPi_{H_{y}^{(0)}}v_{x}

is discontinuous at most at isolated points for each $x\in Q$ where:

$-$ $H_{y}^{(0)}$ is the fiber at $y\in Q$ of the singular affine bundle associated to

system (1.2);

$-$ $B(x,r_{v_{x}})\subset Q$ is an euclidean ball of small radius;

$-$ $v_{x}$ a vector of $H_{x}^{(0)}$ ;

$-$ $\varPi_{H_{y}^{(0)}}v_{x}$ the projection of $v_{x}$ on $H_{y}^{(0)}$ .

Moreover, we show how to calculate a continuous and semialgebraic solution of system (1.2).

2 The setting

Let us start by setting some notations and by showing some important preliminary results that will be used to pursue our goal. We shall endow every $\mathbb{R}^{s}$ used here with euclidean norm.

Notation 1: Let $V\subseteq\mathbb{R}^{s}$ be an affine space in $\mathbb{R}^{s}$ and $w\in\mathbb{R}^{s}$ . We denote the projection of $w$ on $V$ (i.e. the point $v\in V$ that makes the euclidean norm of $v-w$ as small as possible) by $\Pi_{V}w$ .

Notation 2: If $x$ $\in Q$ , consider $Q\ni x\longmapsto A(x)\in M_{r,s}(\mathbb{R})$ and $w\in\mathbb{R}^{s}$ . We denote

$\varPi_{1}(x)\,w=\varPi_{\mathrm{\mathrm{Ker}}A(x)^{\perp}}w,\quad\varPi_{2}(x)\,w=\varPi_{\mathrm{Ker}A(x)}w.$

Notation 3: We denote the $i$ -th column of a matrix $A$ by $A_{i}$ .

Lemma 2.1.

Consider

i) $Q\subseteq\mathbb{R}^{n}$ a compact space,

ii) $Q\ni x\longmapsto A(x)\in M_{r,s}(\mathbb{R})$ a matrix valued semialgebraic function,

iii) $\phi:Q\rightarrow\mathbb{R}^{s}$ a map,

and let $Q\ni x\longmapsto p(x)=\varPi_{1}(x)\,\phi(x)$ be the projection of $\phi$ on $\mathrm{Ker}A(x)^{\perp}$ .

If $Q\ni x\longmapsto\phi(x)$ is a semialgebraic function on $Q$ then $Q\ni x\longmapsto p(x)$ is a semialgebraic function on $Q$ .

Proof.

First of all notice that

\mathrm{Ker}A(x)^{\perp}=\mathrm{Span}\{A(x)_{i}^{T}\}_{i\in\{1,\ldots,r\}}=\mathrm{Span}\left\{\text{rows of }A(x)\right\}.

Then consider for a given $I\subseteq\left\{1,\ldots,r\right\}$

K(I)=\{x\in Q:(A(x)_{\,i}^{T})_{i\in I}\text{ is a basis of }\mathrm{Im}A(x)^{T}\}.

The idea we want to pursue is to project the solution $Q\ni x\longmapsto\phi(x)$ on $\mathrm{Ker}A(x)^{\perp}$ and apply the Tarski-Seidenberg theorem, thus concluding that the projection is semialgebraic. Actually, to apply the Tarski-Seidenberg theorem we need that the dimension of the projection space be independent of $x\in Q$ , so we will be localizing the problem on the $K(I)$ . As a matter of fact on $K(I)$ the dimension of $\mathrm{Ker}A(x)^{\perp}=\mathrm{Span}\{A(x)_{i}^{T}\}_{i\in\{1,\ldots,r\}}=\mathrm{Im}A(x)^{T}$ is constant by definition of $K(I)$ . From the semialgebraicity of $Q\ni x\longmapsto A(x)$ it is then trivial to deduce that $K(I)$ is a semialgebraic (possibly empty) subset on $Q$ since:

$\bullet\;$ the linear independence of $(A(x)_{\,i}^{T})_{i\in I}$ can be translated in terms of the minors of $A(x)$ ;

$\bullet\;$ the condition that the vectors of $(A(x)_{\,i}^{T})_{i\in I}$ are generators of $\mathrm{Im}A(x)^{T}$ can be expressed by the following first-order formula

\forall v\in\mathrm{Im}A(x)^{T},\,\exists(\lambda_{\,i}^{T})_{i\in I}\text{ real numbers s.t.}\>v=\underset{i}{\sum}\lambda_{i}A(x)_{\,i}^{T}.

As $\mathrm{Im}A(x)^{T}$ is semialgebraic by definition we get that that condition is semialgebraic

This is the reason why it suffices to prove the lemma for $K(I)$ since if a function is semialgebraic on a finite collection of semialgebraic sets then it is semialgebraic on their union:

$\bullet\;$ if $K(I)=\textrm{\O}$ there is nothing to prove;

$\bullet\;$ if $K(I)\neq\textrm{\O}$ let

\displaystyle\varGamma

\displaystyle(I)=\{(x,\phi(x)):\quad x\in K(I)\}.

$\circ\;$ If $s\geq r$ we define

V=\left\{(x,y):\quad y=A(x)^{T}\begin{pmatrix}\lambda_{1}\\ \vdots\\ \lambda_{r}\end{pmatrix},\;x\in K(I),\>\lambda_{i}\in\mathbb{R}\right\}\subseteq\mathbb{R}^{n}\times\mathbb{R}^{s}

$\circ\;$ If $s<r$ we define

V=\left\{(x,y):\quad y=\widetilde{A}(x)^{T}\begin{pmatrix}\lambda_{1}\\ \vdots\\ \lambda_{\left|I\right|}\end{pmatrix},\;x\in K(I),\>\lambda_{i}\in\mathbb{R}\right\}\subseteq\mathbb{R}^{n}\times\mathbb{R}^{s}

where $\widetilde{A}(x)\in M_{\left|I\right|,s}(\mathbb{R})$ and the columns of $\widetilde{A}(x)^{T}$ form a basis of $\mathrm{Im}A(x)^{T}$ given by $(A(x)_{i}^{T})_{i\in I}$ . In this case we will still write $A(x)$ in place of $\widetilde{A}(x)$ and we will still write $r$ for $\left|I\right|$ .

Let us now factorize $A(x)^{T}$ by QR decomposition: $A(x)^{T}=Q(x)R(x)$ . As a matter of fact if $m\geq n$ every $m\times n$ matrix can be written as the product of a squared orthogonal $m\times m$ matrix $Q$ and a rectangular $m\times n$ matrix $R$ with the blockwise structure

R=\left(\begin{array}[]{c}R_{1}\\ \hline\cr 0\end{array}\right)

where $R_{1}$ is an $n\times n$ upper triangular matrix and $0$ is the $(m-n)\times n$ zero matrix.

The set $\Gamma(I)$ is semialgebraic since it is the graph of a semialgebraic function, hence

\Gamma^{\prime}(I)=\left\{\begin{pmatrix}I_{n}&0\\ 0&Q(x)^{T}\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}:\quad(x,y)\in\Gamma(I)\right\}

is semialgebraic too. In fact, $Q(x)$ has semialgebraic entries since the ones of $A(x)$ are semialgebraic and the QR decomposition can be computed by multiplying iteratively $A(x)$ by appropriate Householder matrices. These matrices are constructed in the following way.

Letting $\left\|\cdot\right\|$ be the euclidean norm, $z=\left(A(x)^{T}\right)_{1}$ , $v=z+\left\|z\right\|e_{1}$ , $\alpha=\mathbin{\frac{\left\|v\right\|^{2}}{2}}$ and $U_{1}=I_{s}-\frac{vv^{T}}{\alpha}$ where $I_{s}$ is the $s\times s$ identity matrix we have that $U_{1}z=-\left\|z\right\|e_{1}$ . Now, repeating the procedure on the minor of $A(x)^{T}$ obtained by eliminating the first row and column (the new $U_{2}$ matrix has the form $\left(\begin{array}[]{cc}I_{1}&0\\ 0&U^{\prime}\end{array}\right)$ where $U^{\prime}$ is calculated as $U_{1}$ mutatis mutandis) we reach the goal. The $Q$ matrix of the QR decomposition is then the transpose of the product of all the $U_{i}$ constructed in the previous way. The result is semialgebraic because in the construction we used only sums, products and square root extractions of semialgebraic functions (as the entries of $A(x)$ are semialgebraic) that are semialgebraic by definition of semialgebraic function.

We next put

V^{\prime}=\left\{\begin{pmatrix}I_{n}&0\\ 0&Q(x)^{T}\end{pmatrix}v:\quad v\in V\right\}

and see that if $z\in V^{\prime}$ then, by the definition of $V^{\prime}$ , we can write

z=(z_{1},\ldots,z_{n+r},\underbrace{0,\ldots,0}_{s-r})^{T}.

It is important to observe that $V^{\prime}$ is obviously a vector space of $\mathbb{R}^{n+s}$ .

Now, recalling the Tarski-Seidenberg theorem¹¹1Tarski-Seidenberg Theorem $Let$ $A$ a semialgebraic subset of $\mathbb{R}^{n+1}$ and $\pi:\mathbb{R}^{n+1}\rightarrow\mathbb{R}^{n}$ , the projection on the first $n$ coordinates. Then $\pi(A)$ is a semialgebraic subset of $R^{n}$ . Corollary If $A$ is a semialgebraic subset of $\mathbb{R}^{n+k}$ , its image by the projection on the space of the first $n$ coordinates is a semialgebraic subset of $\mathbb{R}^{n}$ ., we notice that the projection $\widetilde{\Gamma}^{\prime}(I)$ of $\Gamma^{\prime}(I)$ onto $V^{\prime}$ is semialgebraic. It follows that

\widehat{\Gamma}^{\prime}(I)=\left\{\begin{pmatrix}I_{n}&0\\ 0&Q(x)\end{pmatrix}v,\quad v\in\widetilde{\Gamma}^{\prime}(I)\right\}

is semialgebraic too. By construction, $\widehat{\Gamma}^{\prime}(I)$ is the graph of $\varPi_{1}(x)\,\phi(x),x\in K(I)$ and so the proof of Lemma 2.1 is complete. ∎

Theorem 2.2.

Consider a compact metric space $Q\subseteq\mathbb{R}^{n}$ and the system (1.2):

A\left(x\right)\phi\left(x\right)=\gamma\left(x\right),\quad x\in Q,

for semialgebraic continuous $A$ and $\gamma$ .

The system has a semialgebraic solution $\phi_{0}:Q\rightarrow\mathbb{R}^{s}$ iff, given a solution $\phi_{1}:Q\rightarrow\mathbb{R}^{s}$ of the system,

p(x)=\varPi_{1}(x)\,\phi_{1}(x)\quad is\>semialgebraic.

Proof.

First of all we show that given a solution $\phi_{1}:Q\rightarrow\mathbb{R}^{s}$ of the system, if $Q\ni x\longmapsto p(x)=\varPi_{1}(x)\,\phi_{1}(x)$ is semialgebraic then there exists a semialgebraic solution of the system (1.2). Notice that

\phi_{1}(x)=\varPi_{2}(x)\,\phi_{1}(x)+\varPi_{1}(x)\,\phi_{1}(x),\quad\forall x\in Q.

From this we get that

\gamma(x)=A(x)\phi_{1}(x)=A(x)\varPi_{1}(x)\,\phi_{1}(x),\quad\forall x\in Q

So $Q\ni x\longmapsto p(x)$ is a semialgebraic solution of the system.

Conversely we show that, given a solution $\phi_{1}:Q\rightarrow\mathbb{R}^{s}$ of the system, if the system has a semialgebraic solution $\phi_{0}:Q\rightarrow\mathbb{R}^{s}$ then

Q\ni x\longmapsto p(x)=\varPi_{1}(x)\,\phi_{1}(x)\;\text{is semialgebraic.}

In fact

A(x)(\phi_{0}(x)-\phi_{1}(x))=\gamma\left(x\right)-\gamma\left(x\right)=0,\quad\forall x\in Q.

Decomposing $\phi_{0}(x)$ and $\phi_{1}(x)$ into their components onto $\mathrm{Ker}A(x)$ and $\mathrm{Ker}A(x)^{\perp}$ we get, respectively,

	$\displaystyle\phi_{0}(x)$	$\displaystyle=\varPi_{2}(x)\,\phi_{0}(x)+\varPi_{1}(x)\,\phi_{0}(x),\quad\forall x\in Q,$
	$\displaystyle\phi_{1}(x)$	$\displaystyle=\varPi_{2}(x)\,\phi_{1}(x)+\varPi_{1}(x)\,\phi_{1}(x),\quad\forall x\in Q.$

Hence

A(x)(\varPi_{1}(x)\,\phi_{0}(x)-\varPi_{1}(x)\,\phi_{1}(x))=0,\quad\forall x\in Q,

so that

\varPi_{1}(x)\,\phi_{0}(x)-\varPi_{1}(x)\,\phi_{1}(x)\in\mathrm{Ker}A(x)\bigcap\mathrm{Ker}A(x)^{\perp}=\{0\}.

Therefore

\varPi_{1}(x)\,\phi_{1}(x)=\varPi_{1}(x)\,\phi_{0}(x),

and since $Q\ni x\longmapsto\varPi_{1}(x)\,\phi_{0}(x)$ is semialgebraic by Lemma 2.1, so is $Q\ni x\longmapsto\varPi_{1}(x)\,\phi_{1}(x)$ . ∎

Let us notice that Theorem 2.2 shows also that, given system (1.2), $Q\ni x\longmapsto\varPi_{1}(x)\,\phi_{0}(x)$ is uniquely defined on $Q$ since it is independent of the solution $\phi_{0}$ .

At this point let us consider a singular affine bundle (or bundle for short) (see [1]), meaning a family $\mathcal{H}=(H_{x})_{x\in Q}$ of affine subspaces $H_{x}\subseteq\mathbb{R}^{s}$ , parametrized by the points $x\in Q$ . The affine subspaces

H_{x}=\left\{\lambda\text{$\in$}\mathbb{R}^{s}:A\left(x\right)\lambda=\gamma\left(x\right)\right\},\quad x\in Q

are the fibers of the bundle $\mathcal{H}$ . (Here, we allow the empty set Ø and the whole space $\mathbb{R}^{s}$ as affine subspaces of $\mathbb{R}^{s}$ .)

Now we define $\mathcal{H}^{(k)}$ to be the $k$ -th Glaeser refinement of $\mathcal{H}$ and $\mathcal{H}^{\mathrm{Gl}}$ to be the Glaeser-stable refinement of $\mathcal{H}$ (their fibers will respectively be denoted by $H_{x}^{(k)}$ and $H_{x}^{\mathrm{Gl}},\quad\forall x\in Q$ ). Notice that the projection on the fibers of $\mathcal{H}$ is not linear as the fibers are affine spaces and not vector spaces.

Lemma 2.3.

Consider a compact metric space $(Q,d_{Q}),\>Q\subseteq\mathbb{R}^{n}$ and an $r\times s$ system of linear equations

A\left(x\right)\phi\left(x\right)=\gamma\left(x\right),\quad x\in Q

where for each $x\in\mathbb{R}^{r}$ the entries of

A(x)=(a_{ij}(x))\in M_{r,s}(\mathbb{R})\quad\text{and }\gamma(x)=(\gamma_{i}(x))\in\mathbb{R}^{r}

are themselves semialgebric functions on $\mathbb{R}^{n}$ .

If there is a semialgebraic solution $\phi:Q\rightarrow\mathbb{R}^{s}$ of the system then $\forall x\in Q,\>\forall v_{x}\in H_{x}^{\mathrm{Gl}}$ :

Q\ni y\longmapsto\gamma_{v_{x}}(y)\coloneqq\varPi_{H_{y}^{(0)}}v_{x}\text{ is semialgebraic.}

Proof.

Consider $x\in Q$ and $v_{x}\in H_{x}^{\mathrm{Gl}}$ . We define

\gamma_{v_{x}}(y)\coloneqq\varPi_{H_{y}^{(0)}}v_{x},\quad\forall y\in Q.

By the definition of $H_{y}^{(0)}$ it is true that

H_{y}^{(0)}=\mathrm{Ker}A(y)+\varPi_{1}(y)\,\phi(y),\quad\forall y\in Q,

and from this expression that

\varPi_{H_{y}^{(0)}}v_{x}=\varPi_{1}(y)\,\phi(y)+\varPi_{2}(y)\,v_{x},\quad\forall y\in Q,

Now, $Q\ni y\longmapsto\varPi_{2}(y)\,v_{x}=v_{x}-\varPi_{1}(y)\,v_{x}$ is semialgebraic, as $Q\ni y\longmapsto v_{x}$ is semialgebraic (since it is constant) and so $Q\ni y\longmapsto\varPi_{1}(y)\,v_{x}$ is semialgebraic by Lemma 2.1. In conclusion, $Q\ni y\longmapsto\varPi_{H_{y}^{(0)}}v_{x}$ is semialgebraic, for it is the sum of semialgebraic functions (recall that $Q\ni y\longmapsto\varPi_{1}(y)\,\phi(y)$ is also semialgebraic by Lemma 2.1 since $\phi$ is a semialgebraic solution of system (1.2)). ∎

After this let us introduce a new notion.

Definition 2.4.

Consider a compact metric space $(Q,d),Q\subseteq\mathbb{R}^{n}$ . Given $\mathcal{H}=(H_{x})_{x\in Q}$ whose Glaeser-stable subbundle is denoted by $(H_{x}^{Gl})_{x\in Q}$ , a semialgebric Glaeser-stable bundle associated with the system (1.2) is a family $\widetilde{\mathcal{H}}^{\mathrm{Gl}}=(\widetilde{H}_{x}^{\mathrm{Gl}})_{x\in Q}$ of affine subspaces $\widetilde{H}_{x}^{\mathrm{Gl}}\subseteq\mathbb{R}^{s}$ , parametrized by the points $x\in Q$ , where the fibers $\widetilde{H}_{x}^{\mathrm{Gl}}$ are given by:

	$\displaystyle\widetilde{H}_{x}^{\mathrm{Gl}}$	$\displaystyle=\{v\in H_{x}^{\mathrm{Gl}}:\exists r_{v}\in\mathbb{R}^{+}\text{ s.t. }$
		$\displaystyle\qquad\quad B(x,r_{v})\ni y\longmapsto\gamma_{v}(y)=\varPi_{H_{y}^{(0)}}v\text{ is semialgebraic}\}.$

It is important to notice that $\widetilde{\mathcal{H}}^{\mathrm{Gl}}$ is indeed a bundle, for $\widetilde{H}_{x}^{\mathrm{Gl}}$ is an affine space, for all $x\in Q$ . As a matter of fact:

$-\;$ if $\widetilde{H}_{x}^{\mathrm{Gl}}=\textrm{\O}$ the space is affine as we assume the empty space to be an affine space;

$-\;$ if $\widetilde{H}_{x}^{\mathrm{Gl}}\neq\textrm{\O}$ , given $v_{0},v_{1},v_{2}\in\widetilde{H}_{x}^{\mathrm{Gl}}$ and $\lambda\in\mathbb{R}$ , we have that

(v_{1}-v_{0})+\lambda(v_{2}-v_{0})+v_{0}\in\widetilde{H}_{x}^{\mathrm{Gl}}.

(2.1)

Property (2.1) holds because $H_{x}^{\mathrm{Gl}}$ is an affine space and since $v_{0},v_{1},v_{2}\in\widetilde{H}_{x}^{\mathrm{Gl}}\subseteq H_{x}^{\mathrm{Gl}}$ we have

(v_{1}-v_{0})+\lambda(v_{2}-v_{0})+v_{0}\in H_{x}^{\mathrm{Gl}}.

Moreover

\forall i\in\{0,1,2\},\,\exists r_{v_{i}}\in\mathbb{R}^{+}\text{ s.t. }\;B(x,r_{v_{i}})\ni y\longmapsto\gamma_{i}(y)=\varPi_{H_{y}^{(0)}}v_{i}\text{ is semialgebraic}.

Considering $\overline{r}=\min\{r_{v_{0}},r_{v_{1}},r_{v_{2}}\}$ we get that on $B(x,\overline{r})$

\gamma(y):=\varPi_{H_{y}^{(0)}}((v_{1}-v_{0})+\lambda(v_{2}-v_{0})+v_{0}).

We now consider the orthogonal decomposition of $\gamma(y)$ on to $\mathrm{Ker}A(y)$ and $\mathrm{Ker}A(y)^{\bot}$

\gamma(y)=\varPi_{1}(y)\varPi_{H_{y}^{(0)}}(v_{1}+\lambda(v_{2}-v_{0}))+\varPi_{2}(y)\varPi_{H_{y}^{(0)}}(v_{1}+\lambda(v_{2}-v_{0})).

Recalling then that the projection of a solution of system (1.2) on $\mathrm{Ker}A(y)^{\bot}$ is unique and considering $B(x,\overline{r})\ni y\longmapsto p(y)=\varPi_{1}\gamma_{v_{i}}(y),\quad i=0,1,2,$ gives

\gamma(y)=p(y)+\varPi_{2}(y)(v_{1}+\lambda(v_{2}-v_{0}))

Note that $p$ is semialgebraic on $B(x,\overline{r})$ by Lemma 2.1 and by the uniqueness of $p$ (that is $p(y)=\varPi_{1}(y)\varPi_{H_{y}^{(0)}}v_{0}$ and $\varPi_{H_{y}^{(0)}}v_{0}$ is semialgebraic by hypothesis). Moreover

B(x,\overline{r})\ni y\longmapsto\varPi_{2}(y)\,(v_{1}+\lambda(v_{2}-v_{0}))=(v_{1}+\lambda(v_{2}-v_{0}))-\varPi_{1}(y)\,(v_{1}+\lambda(v_{2}-v_{0}))

is semialgebraic by Lemma 2.1 as $B(x,\overline{r})\ni y\longmapsto v_{1}+\lambda(v_{2}-v_{0})$ is semialgebraic since it is constant.

Remark 2.5.

By Lemma 2.3 it follows that if a semialgebraic solution of system (1.2) exists then $\widetilde{\mathcal{H}}^{\mathrm{Gl}}=\mathcal{H}^{\mathrm{Gl}}$ . Moreover if for an $x\in Q$ there is a $v_{x}\in H_{x}^{\mathrm{Gl}}$ such that $B(x,r_{v_{x}})\ni y\longmapsto\gamma_{v_{x}}(y)=\varPi_{H_{y}^{(0)}}v_{x}$ is semialgebraic then $\widetilde{H}_{x}^{\mathrm{Gl}}=H_{x}^{\mathrm{Gl}}$ since if $v^{\prime}\in H_{x}^{\mathrm{Gl}}$ then

\begin{array}[]{ccc}B(x,r_{v_{x}})\ni y\longmapsto\varPi_{H_{y}^{(0)}}v^{\prime}&=&\varPi_{1}(y)\,\varPi_{H_{y}^{(0)}}v^{\prime}+\varPi_{2}(y)\,\varPi_{H_{y}^{(0)}}v^{\prime}\\ &=&\varPi_{1}(y)\,\gamma_{v_{x}}(y)+\varPi_{2}(y)\,v^{\prime}\\ &=&\varPi_{1}(y)\,\gamma_{v_{x}}(y)+v^{\prime}-\varPi_{1}(y)\,v^{\prime}\end{array}

where the equality in the second line is due to the uniqueness of the projection of a solution and to the fact that $H_{y}^{(0)}$ is parallel to $\mathrm{\mathrm{Ker}}A(y)$ .

Therefore $B(x,r_{v_{x}})\ni y\longmapsto\varPi_{H_{y}^{(0)}}v^{\prime}=\varPi_{1}(y)\,\varPi_{H_{y}^{(0)}}v^{\prime}+\varPi_{2}(y)\,\varPi_{H_{y}^{(0)}}v^{\prime}$ is semialgebraic as it is a sum of semialgebraic functions by Lemma 2.1 and thus $v_{x}\in\widetilde{H}_{x}^{\mathrm{Gl}}$ .

3 Existence of a continuous semialgebraic solution

Theorem 3.1.

Consider a compact metric space $Q\subseteq\mathbb{R}^{n}$ and a system of linear equations

A\left(x\right)\phi\left(x\right)=\gamma\left(x\right),\quad x\in Q

(3.1)

where the entries of

A(x)=(a_{ij}(x_{1},\ldots,x_{n}))\in M_{r,s}(\mathbb{R})\quad\text{and}\quad\gamma(x)=\left(\gamma_{i}(x)\right)\in\mathbb{R}^{r}

are themselves semialgebric functions on $\mathbb{R}^{n}$ .

Assume that for every given $x\in Q$ such that $\widetilde{H}_{x}^{\mathrm{Gl}}\neq\emptyset$ there exists $v_{x}\in\widetilde{H}_{x}^{\mathrm{Gl}}$ such that $B(x,r_{v_{x}})\ni y\longmapsto\gamma_{v_{x}}(y)=\varPi_{H_{y}^{(0)}}v_{x}$ is discontinuous, at most, at isolated points (that therefore must be finitely many). Then system (3.1) has a continuous semialgebraic solution $\phi:Q\rightarrow\mathbb{R}^{s}$ iff $\widetilde{\mathcal{H}}^{\mathrm{Gl}}$ has no empty fiber.

Proof.

$\bullet\;$ At first we show that if $\widetilde{H}_{x}^{\mathrm{Gl}}$ has no empty fiber then the system (1.2) has a continuous semialgebraic solution $\phi:Q\rightarrow\mathbb{R}^{s}$ .

By hypothesis we have that $\forall x\in Q,\>\exists v_{x}\in H_{x}^{\mathrm{Gl}}\neq\textrm{\O, so that }\exists r_{v_{x}}\in\mathbb{R}^{+}$ s.t. $B(x,\overline{r}_{v_{x}})\ni y\longmapsto\gamma_{v_{x}}(y)=\varPi_{H_{y}^{(0)}}v_{x}$ is semialgebraic and possibly discontinuous at isolated points.

We next claim that

	$\displaystyle\forall x$	$\displaystyle\in Q,\>\forall v_{x}\in\widetilde{H}_{x}^{\mathrm{Gl}}\neq\textrm{\O},\exists\overline{r}_{v_{x}}\text{ with }0<\overline{r}_{v_{x}}<r_{v_{x}}\>\text{ s.t.}$		(3.2)
		$\displaystyle\quad B(x,\overline{r}_{v_{x}})\ni y\longmapsto\gamma_{v_{x}}(y)=\varPi_{H_{y}^{(0)}}v_{x}\text{ is continuous}.$

To prove the claim, let us, by contradiction, assume that

	$\displaystyle\exists x$	$\displaystyle\in Q,\>\exists v_{x}\in\widetilde{H}_{x}^{\mathrm{Gl}}:\forall n\in\mathbb{N},\,\exists y_{n}\in B(x,\frac{r_{v_{x}}}{n})\text{ s.t. }$
		$\displaystyle\quad B(x,r_{v_{x}})\ni y\longmapsto\gamma_{v_{x}}(y)\text{ is discontinuous at }y_{n}.$

We have two cases:

1. $\forall n\in\mathbb{N}\text{ one has }y_{n}\neq x$ . Then $\gamma_{v_{x}}(y)$ has infinitely many isolated discontinuity points which is impossible because $\gamma_{v_{x}}$ is semialgebraic;

2. $\exists\overline{n}\in\mathbb{N}\text{ such that }y_{\overline{n}}=x$ . If $v_{x}\notin H_{x}^{\mathrm{Gl}}$ then $v_{x}\notin\widetilde{H}_{x}^{\mathrm{Gl}}$ since $H_{x}^{\mathrm{Gl}}\supseteq\widetilde{H}_{x}^{\mathrm{Gl}}$ : this is clearly not possible as we took $v_{x}\in\widetilde{H}_{x}^{\mathrm{Gl}}$ . Now, if we had $v_{x}\in H_{x}^{\mathrm{Gl}}$ then on the one hand

\mathrm{dist}(v_{x};H_{y}^{\mathrm{Gl}})\underset{{\scriptstyle y\rightarrow x}}{\longrightarrow}0,

and, on the other,

\left\|\Pi_{H_{y}^{(0)}}v_{x}-v_{x}\right\|=dist(v_{x},H_{y}^{(0)})\underset{{\scriptstyle H_{y}^{\mathrm{Gl}}\subseteq H_{y}^{(0)}}}{\underbrace{\leq}}d(v_{x},H_{y}^{\mathrm{Gl}}).

Therefore

\Pi_{H_{y}^{(0)}}v_{x}\underset{{\scriptstyle y\rightarrow x}}{\longrightarrow}v_{x}\underset{\overset{\uparrow}{{\scriptstyle v_{x}\in H_{x}^{\mathrm{Gl}}\subseteq H_{x}^{(0)}}}}{=}\Pi_{H_{x}^{(0)}}v_{x}.

This is impossible since $\gamma_{v_{x}}(y)$ would be continuous at $x$ , contrary to the assumption. Thus (3.2) holds.

Now notice that the set of balls $\{B(x,\overline{r}_{v_{x}})\}_{x\in Q}$ , where $v_{x}$ is chosen in $\widetilde{H}_{x}^{\mathrm{Gl}}$ , is an open cover of the compact space $Q$ . Then there is $N$ such that $\{B(x_{i},\overline{r}_{v_{x_{i}}})\}_{i=1,\ldots,N}$ is an open cover of $Q$ . Consider

\tau_{(x,r)}(y)=\begin{cases}\sqrt{r^{2}-\left\|y-x\right\|^{2}}&\text{for }y\in B(x,r),\\ 0&\text{for }y\notin B(x,r).\end{cases}

Notice that $\tau_{(x,r)}(y)$ is semialgebraic and continuous on $Q$ , $\forall x\in Q$ , $\forall r\in\mathbb{R}^{+}$ and that ${\displaystyle\sum_{i=1}^{N}}\tau_{(x_{i},\overline{r}_{v_{x_{i}}})}(y)>0$ for each $y\in Q$ as $\tau_{(x,r)}(y)\geq 0$ for every $y\in Q$ and $\tau_{(x,r)}(y)>0$ for all $y\in B(x,r)$ . Moreover, $\forall y\in Q,\,\exists B(x_{i},\overline{r}_{v_{x_{i}}})\text{ as above s.t. }y\in B(x_{i},\overline{r}_{v_{x_{i}}})$ since $\{B(x_{i},\overline{r}_{v_{x_{i}}})\}_{i=1,\ldots,N}$ is an open covering of $Q$ . Hence the function

\phi(y)=\frac{1}{{\displaystyle\sum_{i=1}^{N}}\tau_{(x_{i},\overline{r}_{v_{x_{i}}})}(y)}\sum_{j=1}^{N}\tau_{(x_{j},\overline{r}_{v_{x_{j}}})}(y)\Pi_{H_{y}^{(0)}}v_{x_{j}}

is a semialgebraic and continuous solution of the system on $Q$ .

$\bullet\;$ Conversely, we show that if system (1.2) has a continuous semialgebraic solution $\phi:Q\rightarrow\mathbb{R}^{s}$ then $\widetilde{H}_{x}^{\mathrm{Gl}}$ has no empty fiber.

By Remark 2.5 we have that $\widetilde{\mathcal{H}}^{\mathrm{Gl}}=\mathcal{H}^{\mathrm{Gl}}$ but $\mathcal{H}^{\mathrm{Gl}}$ has no empty fiber because there is a continuous solution of the system (1.2) as shown in [1].

The proof of the theorem is complete. ∎

Theorem 3.1 gives therefore an answer to the initial problem of determining a necessary and sufficient condition for the existence of a solution $Q\ni x\longmapsto\phi\left(x\right)=\left[\begin{array}[]{c}\phi_{1}(x)\\ \vdots\\ \phi_{s}(x)\end{array}\right]\in\mathbb{R}^{s}$ of the system (1.2), with the $\phi_{i}:Q\rightarrow\mathbb{R}$ continuous and semialgebric.

Remark 3.2.

Let $p$ be the projection of a solution of the system (1.2) on $\mathrm{Ker}A(y)^{\perp},\,y\in Q$ . If $p$ is not semialgebraic the system has no semialgebraic solution by Theorem 2.2 and so it has no continuous and semialgebraic solution. Otherwise, if $p$ is semialgebraic then $\widetilde{\mathcal{H}}^{\mathrm{Gl}}=\mathcal{H}^{\mathrm{Gl}}$ since if $v\in H_{x}^{\mathrm{Gl}}$ we may write

Q\ni y\longmapsto\varPi_{H_{y}^{(0)}}v=p(y)+\varPi_{2}(y)\,v=p(y)+v-\varPi_{1}(y)\,v

that is semialgebraic by Lemma 2.1. We have that $\gamma_{v_{x}}(y)=p(y)+\varPi_{1}(y)\,v_{x}$ , $\forall y\in B(x,r_{v_{x}})$ since $\gamma_{v_{x}}(y)$ is the projection of $v_{x}$ on $H_{y}^{(0)}$ . This implies that for each $x\in Q$ there is a $v_{x}$ such that $\gamma_{v_{x}}$ is semialgebraic iff $p$ is semialgebraic and $\gamma_{v_{x}}$ is discontinuous at most at isolated points for each $x\in Q$ iff $p$ and $B(x,r_{v_{x}})\ni y\longmapsto\varPi_{1}(y)\,v_{x}$ are discontinuous at most at isolated points for each $x\in Q$ . Note that if there is a $v_{x}$ such that $\gamma_{v_{x}}$ is semialgebraic then $\gamma_{v_{x}}$ is semialgebraic for all $v\in H^{\mathrm{Gl}}$ by Remark 2.5.

Hence, if we know a solution $\phi$ of system (1.2) or, at least, its projection on $\mathrm{Ker}A(x)^{\perp}$ , Theorem 3.1 can be written in the following (equivalent) form.

Theorem 3.3.

Let $p$ be the projection of a solution of the system on $\mathrm{Ker}A(x)^{\perp}$ and assume that for every given $x\in Q$ there exists $v_{x}\in\widetilde{H}_{x}^{\mathrm{Gl}}$ such that $B(x,r_{v_{x}})\ni y\longmapsto\gamma_{v_{x}}(y)=\varPi_{H_{y}^{(0)}}v_{x}$ is discontinuous, at most, at isolated points (which is the same as saying that $p$ and $B(x,r_{v_{x}})\ni y\longmapsto\varPi_{1}(y)\,v_{x}$ are discontinuous at most at isolated points for every given $x\in Q$ ). We have that the following conditions are equivalent:

i) The system has a continuous and semialgebraic solution.

ii) $\widetilde{\mathcal{H}}^{\mathrm{Gl}}$ has no empty fiber.

iii) The Glaeser-stable refinement $\mathcal{H}$ associated with the system has no empty fibers and $p$ is semialgebraic.

iv) The system has a continuous solution and a semialgebraic one (they may possibly be different).

Proof.

We showed that $i)\Leftrightarrow ii)$ in Theorem 3.1 (using also Remark 3.2). The proof of $ii)\Leftrightarrow iii)$ follows from Remark 3.2 and that of $iii)\Longleftrightarrow iv)$ from Fefferman’s result that a system like (1.2) has a continuous solution iff the Glaeser-stable refinement $\mathcal{H}$ associated with the system has no empty fibers (see [1]) and from Theorem 2.2. ∎

References

[1] C. Fefferman, J. Kollár, Continuous Solutions of Linear Equations, From Fourier analysis and number theory to Radon transforms and geometry, Dev. Math., vol. 28, Springer, New York, 2013, pp. 233-282. MR 2986959

Semialgebraic Solution of Linear Equation with Continuous Semialgebraic Coefficients

1 Introduction

2 The setting

Lemma 2.1.

Proof.

Theorem 2.2.

Proof.

Lemma 2.3.

Proof.

Definition 2.4.

Remark 2.5.

3 Existence of a continuous semialgebraic solution

Theorem 3.1.

Proof.

Remark 3.2.

Theorem 3.3.

Proof.

References

Semialgebraic Solution
of Linear Equation with
Continuous Semialgebraic Coefficients