14 problems

Let $\mathcal{F}$ be the family of polynomial systems $F$ such that $Z(F)$ is finite and the Galois group of $F$ over $\mathbb{Q}$ is dihedral or bicyclic. Let…

Fewnomial root-count conjecture. The maximum number of non-degenerate roots of $F$ in the positive orthant is

Consider a chemical reaction network (CRN) whose steady-state ideal satisfies Assumptions 1–3: mass-action kinetics, inclusion of conservation laws, and absence of independent subn…

Let $n$ be the dimension and $d$ the degree. Consider a system of $n$ degree-$d$ Chebyshev polynomials in $n$ variables, and suppose that $d$ is sufficiently large. Second-step com…

Zeng's deflation conjecture. The recursive deflation process terminates after finitely many steps, the deflation sequences of all points $\textbf{x}\in U$ are equal and regular, an…

Let $M=2s$, and let $P_a^M(U_i,\bar U_i)=0$ be the polynomial Bethe-root equations for parameter choice $a=d$ or $a=g$. A solution is admissible if its coordinates are pairwise dis…

Let $F(\mathbf{z})=G(\mathbf{z})/H(\mathbf{z})$ be a rational function, and let Assumption (J2) denote the Jacobian nondegeneracy condition used in the paper's critical-point syste…

Let $A$ be an irreducible tuple, and let the three discriminant sets be those of Definition 2.33, consisting of the discriminant notions defined earlier in the paper. Coincidence c…

Let $A$ be a tuple in the setting of Step (1) of Classification, and let $B$ be reduced and irreducible of mixed volume $d$. Write $j$ for the map occurring in that classification,…

Let $F=\{f_1,\ldots,f_k\}$ be a system of tropical polynomials in $n$ variables, and let $M_N$ be the finite Macaulay matrix obtained from the monomial multiples of the $f_i$ of de…

Let $A=(a_{i,j})_{1\leq i,j\leq n}$ be a complex invertible matrix, and let $x_1,\dots,x_n$ be complex variables. Nonvanishing solution conjecture. The system … has a solution…

Let $A_1,A_2,\ldots,A_n\subset\mathbb{Z}^n$ be a reduced tuple, meaning that the dimension of the convex hull of $\bigcup_{j\in I}A_j$ is greater than $|I|$ for every…

Let $OmegaL(n,k)$ denote the maximum number of non-degenerate roots in a local field $L$ of a system of $n$ polynomials in $n$ variables having at most $k$ monomials in total. The…

Minimum-energy conjecture. For every $R\geq 0$,