9 problems

Gyarmati–Ruzsa conjecture. The equation $x^2+y^2=2z^2$ is partition regular.

Let $X$ be a smooth, toric, projectively normal variety. Sturmfels's quadratic-generation conjecture. The variety $X$ is defined by quadrics, equivalently its homogeneous ideal is…

Ergodic variant of the quadratic Rado conjecture. There exist distinct $x,y,z\in\mathbb{N}$ such that

Quadratic Rado conjecture. The equation

Generalized Pythagorean-pair conjecture. If at least one of $ac$, $bc$, or $(a+b)c$ is a square, then $ax^2+by^2=cz^2$ is partition regular with respect to $x,y$.

Rado triple conjecture. The equation

FKM's pairwise partition-regularity conjecture. If at least one of $ac$, $bc$, or $(a+b)c$ is a square, then $a x^2+b y^2=c z^2$ is partition regular with respect to $x,y$.

Let $A_1,\ldots,A_m\in S^n$ be the symmetric matrices defining the homogeneous quadratic system $(000)$, and let the sufficient conditions in Theorem 4.4 and Corollary 4.5 refer to…

Let $\mathcal{L}_0$ and $\mathcal{L}_1$ be matrices, and consider the quadratic matrix equation $X^2-\mathcal{L}_1X-\mathcal{L}_0=0$. A universal, possibly multivalued matrix funct…