33 problems

Let a fewnomial system in $m$ variables have $|f|$ monomials in total, and consider its real zeros in the positive real quadrant. Kouchnirenko's conjecture. The maximum number of s…

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

Nonzero-solution conjecture. This system admits a solution $(x_1,x_2,\dots,x_n)$ such that $x_i\neq0$ for all $i$. The paper proves a relaxed version of this conjecture, rather tha…

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…

Uniqueness conjecture. The $(n+1)$-by-$n$ polynomial system

Let $K$ be an infinite field and let $R=K[x_1,\ldots,x_n]$. Let $d_1,\ldots,d_m>1$, and let $f_1,\ldots,f_m\in R$ have degrees $d_1,\ldots,d_m$ and no constant terms. Homogenize th…

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 $n$ be fixed, and consider a system of $n$ real polynomial equations in $n$ variables with $t$ distinct exponent vectors. A positive solution is nondegenerate if the Jacobian o…

Let $P$ be a polytope, and let $\mathcal{F}_0=(f_0,\dots,f_s)$ be a polynomial system with Hilbert series … The system is semi-regular when … where $\operatorname{ES}_P(t)$ is the…

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 $M$ be a positive integer, let $P_{sp}^M(U_i,\bar U_i)$ be the polynomial equations defining the Bethe roots, and call a solution $\bar U=\\{U_1,U_2,\ldots,U_M\\}$ admissible w…

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 $F(\mathbf{z})$ be a rational function in $n$ variables, and consider the assumptions on $F$ required to apply the non-combinatorial complexity theorem, apart from the existenc…

Let $q$ be a prime power, let $e_r(d,m)$ denote the maximum number of $\mathbb F_q$-rational common zeros in projective $m$-space of $r$ linearly independent homogeneous polynomial…

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 $f_1,\ldots,f_{n+1}\in K[x_1,\ldots,x_n]$ be generic homogeneous polynomials of degree $d$, so that $c=1$. Let $D^*$ be the smallest degree at which the XL algorithm succeeds,…

Consider decoupled molecules with binding polynomials of bidegree $(n_1,n_2)$, and let the number of solutions mean the number of solutions of the corresponding polynomial system,…

Common-factor conjecture. The maximum number of common zeros in ${\mathbb P}^m({\mathbb F}_q)$ of a system of $r$ linearly independent polynomials in $S_d$ is

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…

Consider the design of an $n$-stage Marx generator electrical network, formulated as a structured system of $n$ polynomial equations in $n$ unknowns with rational coefficients. Rea…