9 problems

Let $d\geq 1$. A homogeneous form of degree $2d$ in three variables is nonnegative if $F(a)\geq 0$ for every $a\in\mathbb{R}^3$, and it admits a positive quadratic representation i…

Buckley–Šivic's conjecture. The Robinson polynomial does not admit a positive quadratic representation. This is the negative answer to the question of whether every nonnegative ter…

Let $P_{n,d}$ be the cone of nonnegative forms of degree $d$ in $n$ variables, let $\Delta_{n,d}$ be the cone of such forms that are not sums of squares, and let…

Let $P_{3,6}$ be the cone of nonnegative ternary sextic forms, let $\Delta_{3,6}$ denote the cone of such forms that are not sums of squares, and let $\mathcal{E}(P_{3,6})$ denote…

Let $\tau$ be a chain type, and let $P^\tau\in\mathbb{Z}[z,w]$ be the polynomial associated with $\tau$ in the paper's recursive construction. Nonnegative-coefficients conjecture.…

Maximal-loss conjecture. In each projection step, one loses the maximal number of quadrics, namely the codimension of the projected variety. Equivalently, successive general-point…

Let $A$ be a support set, and let the sonc cone and nonnegativity cone associated with $A$ be denoted by $\mathcal{S}^+_A$ and $\mathcal{P}^+_A$, respectively. A Chandrasekaran–Mur…

Let $C$ be a nonsingular affine curve over ${\mathbb R}$ with $C({\mathbb R})$ compact, let $V=C\times{\mathbb A}^1$, and let $f\in{\mathbb R}[V]={\mathbb R}[C][y]$ be nonnegative…

Real-point conjecture. For $p \in P_{3,6} \setminus \Sigma_{3,6}$ there exist $v_1,\ldots,v_9 \in \mathbb{R}^3$ yielding a separating extreme ray for $p$. Analogously, for…