341 problems

A convex semialgebraic set is a semialgebraic subset of some finite-dimensional real vector space that is convex, and a spectrahedral shadow is the image of a spectrahedron under a…

Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a semialgebraic spline, meaning a continuous piecewise polynomial function on a semialgebraic partition of $\mathbb{R}^n$. Pierce–Birkhoff…

Let $F=(f_1,\ldots,f_n):\mathbb{R}^n\to\mathbb{R}^n$ be a polynomial map, and let $DF(\mathbf{x})$ denote its Jacobian matrix at $\mathbf{x}$. The real Jacobian conjecture. If … fo…

Lax conjecture. The curve $C$ is hyperbolic with respect to $p$ if and only if it has a determinantal representation by a symmetric matrix $M$ of linear forms such that $M(p)$ is a…

Let $p\in\mathbb{R}[\mathbf{x}]$ be a real-zero (RZ) polynomial with $p(\mathbf{0})=1$. For a polynomial $f$, write $\operatorname{rcs}(f)$ for its rigidly convex set. Generalized…

Let $\mu\geq 1$ and $k\geq\mu+1$ be nonnegative integers, let $f$ be associated with a real Legendre parameter $\lambda$, and fix $\lambda\neq 0,1$. Let $P_{\mu,k}$ be the correspo…

A homogeneous polynomial in $\mathbb{R}[x,y]$ is called hyperbolic if its Hessian polynomial has no real linear factors and is non-positive at every point. For each degree $n$, con…

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a real polynomial map, and let $f'(x)=\det\operatorname{Jac}_x f$. Strong real Jacobian conjecture. If … for every $x\in\mathbb{R}^n$, then…

Let $f:\mathbb{R}^4\to\mathbb{R}^2$ be a polynomial map with an isolated singularity at the origin, meaning that $f(0)=(0,0)$, its first derivatives vanish at the origin, and its J…

$X_9^2$ component conjecture. These representatives exhaust the components of the complement of the discriminant variety; in total there are 52 such components.

Let $X$ be a projective and simply connected M-surface, and write $\mathbb{R}X$ and $\mathbb{C}X$ for its real and complex loci. Let $\beta_1(\mathbb{R}X)$ denote the first Betti n…

Let $\Sigma$ be a ring and let $\alpha,\beta\in\operatorname{Sper}\Sigma$. The connectedness property at $\alpha$ and $\beta$ requires that, for every finite collection…

Let $P_{\mathbb{P}^n,2d}$ be the cone of globally nonnegative forms of degree $2d$ on $\mathbb{P}^n$. Extreme-ray conjecture. Every globally nonnegative form spanning an extreme ra…

Let $f$ and $g$ be the polynomial system in two variables … where $a,b,r,s,t$ are positive reals. Itenberg–Roy conjecture. Such a system has at most $2$ real solutions. Li and Wang…

Let $\operatorname{Gr}(k,n)$ be a Grassmannian, let $(\lambda^1,\dotsc,\lambda^r)$ be a Schubert problem, and let $F^1_{\bullet},\dotsc,F^r_{\bullet}$ be flags secant to a rational…

Let $V$ be the variety under consideration, let its RR discriminant be defined by a polynomial, and call a factor isotropic when it belongs to the isotropic part of that discrimina…

Consider a real polynomial system in two variables … where $f$ has $t\geq 3$ non-zero terms and $g$ has three non-zero terms, with real exponents allowed. Kushnirenko's conjecture.…

Let $f\colon\mathbb R^n\to\mathbb R^n$ be a polynomial map with nowhere zero Jacobian determinant. Let $S_f$ denote the set of points where $f$ is not proper, and write…

Let $d\geq 2$ and $n\geq 2$. A real morphism $f:\mathbb{P}^n\rightarrow\mathbb{P}^n$ is a morphism defined over $\mathbb{R}$, where $\mathbb{P}^n$ is complex projective $n$-space.…

Let $(X,\mathcal E)$ be a compact non-singular real tropical hypersurface. Let ${\mathcal K}_1,\dots,{\mathcal K}_n$ and ${\mathcal S}_{\mathcal E}$ be the coefficient systems appe…

Let $h\in\mathbb{R}[x_1,\dots,x_n]$ be hyperbolic with respect to $\mathbf{e}=(e_1,\dots,e_n)\in\mathbb{R}^n$, and let $\Lambda_{++}(h,\mathbf{e})$ be its open hyperbolicity cone.…

Let a general system consist of $m$ polynomials in $m$ variables, and measure its sparsity by the number of nonzero terms appearing among the polynomials. Kushnirenko's conjecture.…

Let $\ell_1,\dotsc,\ell_4$ be lines tangent to a twisted cubic at real points. Shapiro conjecture. The two solutions to the corresponding problem of four lines are real. This is th…

Nash's rationality conjecture. There exists a rational variety $X$ whose real locus is diffeomorphic to $M$:

A semi-algebraic set is a subset of $\mathbb R^n$ definable in the first-order structure $(\mathbb R;+,*)$. A set is semidefinite representable when it is primitive positive defina…