26 problems

Shiota's conjecture. ${\EuScript S}$ is a Nash image of ${\mathbb R}^d$ if and only if ${\EuScript S}$ is pure dimensional and there exists an analytic path

Let $f$ be a semialgebraic function with a stratification of its graph, let $\cX$ denote the relevant stratum, and let $\partial\cX$ and $\sX_{\dim(\cX)-1}$ denote its boundary and…

A semialgebraic set $\EuScript T\subset\mathbb R^n$ is locally connected by analytic paths if, for every $x\in\EuScript T$ and every open semialgebraic neighborhood…

Let $K$ be a finite subset of $cmathbb{R}^{n\times m}$, and let $\operatorname{conv}_\mathrm{rc}K$ denote its rank-one convex hull, namely the set of matrices satisfying…

Let $U \in \mathcal{U}(n,m)$, and let … be its unique partition into irreducible sets. Let $\gamma \in C_{\operatorname{cv}}(U) \cap \mathbb{R}_{>0}^m$, and define … Here…

Higher-uniformity Zarankiewicz conjecture. For $k>2$, the exponents of $u$ and $D$ in the semialgebraic Zarankiewicz bound can be decreased so that the bound is tight in the sense…

Equitable regularity conjecture. The equitable semialgebraic regularity lemma holds with partitions into

Let $P$ and $Q$ be polynomial functors over $\mathbb{R}$, let $Y\subseteq Q$ be a semialgebraic subset, and let $\alpha:Q\to P$ be a polynomial transformation. Write…

Fix $m\ge 0$. Let $(A_{ij}(x))_{1\le i\le N,1\le j\le M}$ be a matrix of semialgebraic functions on $\mathbb{R}^n$, and let the system … L{mu,nu}f(x)=sum{i=1}^Nsum{|alpha|le bar m}…

Finite-convergence conjecture. If some classical optimality conditions hold at every global minimizer of the polynomial optimization problem, then

Adaptive-radius homology computation conjecture. One can compute the homology of

Let $W\subset\mathbb R^N$ be a compact body with smooth semialgebraic boundary, and let a lacuna be a connected domain in the space of affine hyperplanes, consisting of hyperplanes…

Link and double-point image criterion. The map germs $f$ and $g$ are bi-Lipschitz equivalent if and only if $link(f)$ and $link(g)$ are bi-Lipschitz equivalent and the images…

Let $\mathcal P$ be a strong cylindrical algebraic decomposition of $\mathbb R^n$ adapted to a closed and bounded semi-algebraic set $S$. Lazard's conjecture. The decomposition ind…

Let $(S_0,0)$ and $(S_1,0)$ be two normally embedded real semialgebraic surface germs. They are normally embedded when their inner and outer metrics are bi-Lipschitz equivalent; fo…

Let $X$ be an $\mathscr{A{\!}R}$-closed set in $\mathbb{R}^n$, and let $\mathscr{A}_a(X)$ denote the ring of arc-analytic semialgebraic functions on $X$. Extension conjecture. Ever…

Let $X$ be an $\mathscr{A{\!}R}$-closed set in $\mathbb{R}^n$. Denote by $\mathscr{A}_a(X)$ the ring of arc-analytic semialgebraic functions on $X$, by…

Rank-one factor abundance conjecture. Every such matrix has a psd factorization in which either at least $k$ matrices $A_i$ and $k-1$ matrices $B_j$ are rank one, or at least $k-1$…

Semialgebraicity conjecture. The feasible TDOA set $\Theta_n$ can be described by algebraic equations and inequalities; equivalently, $\Theta_n$ is a semialgebraic variety.

Let $X$ be the state space and let $(\rho^k)$ be the sequence of discounted occupation-measure densities used in the paper's approximation scheme. The existing assumption requires…

Let $A$ be a semialgebraic set inside the unit cube $I^n\subset\mathbb R^n$, and let $D(A)$ denote its diagram. Let $C(\epsilon,A)$ be the analytic approximation complexity at accu…

Let a spectrahedral shadow be the affine projection of a spectrahedron, that is, a set of the form … where $F_k \in {\mathbb S}^m$ for $k=0,1,\ldots,n$ and $G_l \in {\mathbb S}^m$…

Basic closed semialgebraic conjecture. The set $\cS$ is a basic closed semialgebraic subset of $H'$.

Semialgebraic universal-cover classification conjecture. The following are equivalent:

Let $\mathcal{C}_{\mathcal{G}}$ denote the cone associated with a graph $\mathcal{G}$. A basic open semialgebraic set is an open semialgebraic set defined by finitely many strict p…