178 problems

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.

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…

Let $Q_1,Q_2,Q_3\subseteq\mathbb{R}^2$ be three general real conics. A real circle is a circle defined over $\mathbb{R}$ that is tangent to each of $Q_1,Q_2,Q_3$. Maximality conjec…

Polynomial growth conjecture. For any fixed $p\geq 0$, $m_{p,\{\lambda\}_n}(V_n(I))$ is eventually given by a polynomial in $n$.

Let $A$ be a regular ring, let $\alpha,\beta\in\operatorname{Sper} A$, and let $g_1,\dots,g_s$ be a finite collection of elements of $A$ not belonging to…

Let $m,p\geq 1$, let $\gamma$ be the real rational normal curve, and let $w_1,\ldots,w_n$ be Schubert conditions satisfying … For distinct real points $s_1,\ldots,s_n\in\gamma$, co…

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 $C\subset\mathbb{C}^2$ be a curve defined over $\mathbb{Z}$ and irreducible over $\mathbb{C}$. Let $C_{\mathbb{R}}$ be an irreducible component of $C\cap\mathbb{R}^2$ whose int…

Let $m,p,n$ be positive integers with $n=mp$. Let $(\bar{A},\bar{B},\bar{C})$ be a minimal realization of the system represented through a coprime factorization $D^{-1}(s)N(s)$, wh…

Let $X$ be a smooth projective rational surface, and let $n$ be a natural integer. The group of algebraic diffeomorphisms $\operatorname{Diff}_{\mathrm{alg}}(X(\mathbb{R}))$ consis…

Let ${\cal G}$ be an $\mathbb R$-rational building set. A real subspace $C\in{\cal C}_{\cal G}$ is purely complex when it decomposes as … Write $\mathbb R({\cal G})$ for the real s…

Let $0\le d<n$, let $t_1<t_2<\dots<t_{(n+1)(d-n)}$ be real numbers, and let $\mathbf r:\mathbb C\to\mathbb C^{d+1}$ be a rational normal curve with coordinates $r_i(t)=t^i$ for…

Separation conjecture. If g\text{in \hspace{-.8em}/}\langle\alpha,\beta\rangle, then $\alpha$ and $\beta$ lie in the same connected component of

Pierce–Birkhoff conjecture, abstract version. For every piecewise-polynomial function $f$ and every $\alpha,\beta\in\operatorname{Sper} A$, if $f_\alpha$ and $f_\beta$ are local re…

The $G^1$-degeneration conjecture. The spectral sequence $G^k_{p,q}$ for $H_*(X(\mathbb{R}))$ degenerates at the $G^1$ level.

A real algebraic variety is a complex variety $X$ defined over the real numbers, with real part $X(\mathbb{R})$ equal to the fixed-point set of complex conjugation. It is an M-vari…

Let $X$ be a smooth real projective variety. Let $RT_tH_n(X)$ denote the topological filtration on $H_n(ReX;\mathbb Z_2)$, given by the image of $\Phi_{t,n}$, and let $RG_tH_n(X)$…

Let $(w_1,\dotsc,w_m)$ be a Grassmannian Schubert problem for $\mathbb{F}\ell(\alpha;n)$, let $\delta(w_i)$ be the unique descent of $w_i$, and let $\Delta_w(t_1,\dotsc,t_m)$ be it…

Let $F(x,y)$ be a polynomial whose zero set is a plane curve consisting of the product of $d$ real lines. Consider the critical points of $F$ having the same non-zero critical valu…

Assume that $\mathbb{P}^2_k$ is obtained from $\mathbb{P}^2$ by blowing up $k$ generic real points and is equipped with its natural real structure. Let $D\subset \mathbb{P}^2_k$ be…

Let a simple complex manifold mean a complex manifold satisfying the source’s informal notion of simplicity, and likewise let a simple real form mean a real form satisfying the cor…

Let $M(\mathbb C)$ be a smooth projective manifold. A real form of $M(\mathbb C)$ is a real algebraic manifold whose complexification is $M(\mathbb C)$. Finiteness conjecture. The…