16 problems

Let $R(k)=\frac{3k^2-3k}{2}$. Let $\mathcal{C}$ be a non-singular real algebraic curve of degree $2k$ in $\mathbb{P}^2$, with $p$ even ovals and $n$ odd ovals. Ragsdale's conjectur…

Let $n \geq 3$ be an odd integer and let $X \subset \mathbb{P}^{n}$ be an unramified real curve. A branch is a connected component of the real locus $X(\mathbb{R})$. Huisman's conj…

Let $C$ be a smooth real plane curve, and call an oval empty when its interior contains no other real component. Itenberg–Viro contraction conjecture. Any empty oval of $C$ can be…

Let $A$ be an $M$-curve with a deep nest, meaning a nest with at least three levels of nesting, of arbitrary degree. Orevkov's conjecture. $A$ must have some empty outer ovals. Thi…

Rokhlin's conjecture. Both Arnold surfaces ${\mathfrak A}_+$ and ${\mathfrak A}_-$ are standard.

Let $6\cdot\Delta_2$ denote the dilation by $6$ of the standard two-dimensional simplex, and let a regular unimodular triangulation of $6\cdot\Delta_2$ be a triangulation induced b…

Let $R(k)=\frac{3k^2-3k}{2}$. For a non-singular real algebraic curve $\mathcal{C}$ of degree $2k$ in $\mathbb{P}^2$, let $p$ and $n$ denote the numbers of even and odd ovals, resp…

21-diameter conjecture. The maximal number of real diameters obtainable by a small deformation of three real lines is $21$.

Diameter upper-bound conjecture. The number of diameters of $\mathcal R$ does not exceed

Curvature-extrema conjecture. The number of minima of the curvature on $\mathcal R$ plus the number of inflection points on $\mathcal R$ equals the number of bounded edges of…

Asympture conjecture. The first terms of the sequence suggest that $a'_0=\alpha$ is sufficient; under this choice, the exponential-growth constant $\rho$ in the bound for the numbe…

Let a smooth real plane curve be of even degree, so that an oval is null-homotopic in $\mathbb RP^2$. Call a scheme rigid when any two curves representing it are rigid-isotopic. On…

Let $C_m$ be any smooth real plane curve of degree $m$, and let its empty ovals be the components whose interiors contain no other real components. Collective contraction conjectur…

Fix an integer $m\ge 1$, and consider real schemes of degree $m$, ordered in the lattice of all real schemes. A scheme is of type I when its real curve is dividing. Rohlin's maxima…

Totally real pencil conjecture. Any curve whose real scheme is of type I may be endowed with a totally real pencil.

Let $C_9$ be an $M$-curve of degree $9$ with real scheme … Here, the integers $\alpha_i$ count the empty ovals in the three nests. Korchagin's conjecture. At least two of the…