184 problems

For every $n\ge 1$, real coefficients $c_1,\ldots,c_n$, and polynomials $P_1,\ldots,P_n\in\mathbb{R}[x]$ satisfying $\deg P_k\le 2$ and $P_k(x)>0$ for all $x\in\mathbb{R}$, if…

Let $F$ be a generic homogeneous form, and let $\mathcal{R}_F$ denote the factor of the resultant appearing in Theorem. Irreducibility conjecture. The polynomial $\mathcal{R}_F$ is…

Let $\mathbb{R}\langle \xi\rangle$ be the real closed field of algebraic Puiseux series, let $\mathcal{P}=\{P_1,\ldots,P_s\}\subset \mathbb{R}[X_1,\ldots,X_n]$, and set … For a bou…

Nondifferentiability conjecture. If $f$ is non-differentiable at infinitely many points, then $C$ is not semi-algebraic.

Consider a convex multi-objective optimization problem $\min (f_1(x),\ldots,f_m(x))$ over $X=\{x\in\mathbb{R}^n:g_j(x)\le 0\ \text{for }j=1,\ldots,r\}$, where the objective functio…

Let $A\in\mathbb{R}_{\geq 0}^{M\times N}$ have non-negative rank $H_0$, and suppose that $A=U_0V_0$ for some strictly positive matrices $U_0\in\mathbb{R}_{>0}^{M\times H_0}$ and…

The supplied sources identify Ebenfelt's weak SOS conjecture as a rank-gap assertion for sums of squares of Hermitian polynomials, but they do not state the precise hypotheses, def…

Determine the maximum number of distinct real circles tangent to each of three general real conics in the Euclidean plane. The proposed conjecture was that every such triple has at…

For every $n\in\mathbb{N}$ and every Hermitian matrix-valued Laurent polynomial $P(z,w)=\sum_{(j,k)\in F}A_{j,k}z^jw^k\in M_n\!\left(\mathbb{C}[z^{\pm1},w^{\pm1}]\right)$ satisfyin…

Consider a system of $n$ polynomials in $n$ variables, where the $i$th polynomial has $m_i$ monomials and all solutions are simple. Kouchnirenko's conjecture. The system has at mos…

Monotonicity conjecture. For any $n\in\mathbf{N}$, we have

Let $t_2$ denote the number of crossing points of multiplicity $2$ in a real line arrangement, and let $s$ be the number of its lines. Dirac–Motzkin conjecture. The inequality … ho…

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 $m,p\geq 1$, let $\gamma$ be the real rational normal curve, and let $w_1,\ldots,w_n$ be Schubert conditions on $p$-planes in $(m+p)$-space. For points…

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 $V$ 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…

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…

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

Let $f:\mathbb{R}^4\to\mathbb{R}^2$ 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…

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…

Let $\Sigma$ be a ring. A piecewise-polynomial function on $\operatorname{Sper}\Sigma$ is an element of $PW(\Sigma)$, and $\Sigma$ is a Pierce–Birkhoff ring if every…

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

Let … where $0<\gamma_{2n-1}<\cdots<\gamma_1$, and let $S(x),P(x)\in\mathbb{R}[x]$ satisfy the relation … with $P(0)\neq 0$. Reality conjecture for $S$. The polynomial $S(x)$ has o…

Let $f_1,\ldots,f_t$ be relatively prime real polynomials of positive degrees with respect to $x_{f+1},\ldots,x_r$, and let $x=(x_1,\ldots,x_f)$. Let $D(x)$ be a generalized discri…

Let $f\colon\mathbb{R}^2\to\mathbb{R}$ be a Morse polynomial of degree $4$, and consider its principal part. The principal part may be non-discriminant positive definite, vanish on…