45 problems

High-accuracy conjecture. High accuracy should still be achievable over the field of rational numbers when

Strong and weak conjectures. The strong conjecture asserts that all essential specializations are singular. The weak conjecture asserts

Let $k$ be a finite field with $|k|=q$, let $X$ be a set of state transition pairs, and let $M_X$ be the associated monomial ideal with generating set $W$. A monomial tree is the d…

Minimal-resolution conjecture. In the homogenized Weyl algebra, a minimal resolution exists and can be constructed by the algorithm in LaScala, since the algebra is graded; however…

Completeness conjecture. There are no other nonassociative Moufang loops of order $64$.

Let $\Gamma_i=(C_{i1},\dots,C_{is_i})$ be the set of all segments of a disjoint reduced CGS whose reduced bases have a common leading power product $\operatorname{lpp}_i$. A common…

Let $k$ be a field, and let $\omega$ be the smallest exponent such that linear algebra on square $n\times n$ matrices over $k$ has complexity $O(n^{\omega+\epsilon})$ for every…

Let $I$ be a reduced equidimensional homogeneous ideal, and let $m(I)$ denote its regularity. In the preceding bound, part (a) asserts the linear estimate $m(I)\leq \deg Z-n+r+1$ u…

Let $I$ be a homogeneous ideal in a polynomial ring in $n$ variables. Write $d(I)$ for the maximum degree of a minimal set of generators of $I$, and $m(I)$ for the regularity of…

Let $P$ be the degree-246 polynomial arising from the elimination for the degree-2 absolute equiripple approximation of $(1+t^3)^{1/3}$ on $[0,1]$. Let $2^{1/3}$ be the real cube r…

For $n,m,l\ge 1$, let $c_{n,m}$ denote the constants arising from the cyclic case, and let $G_d$ be the set of relations defined from the displayed relations for…

Quadratic-rank conjecture. The linear system suffices for tensor decompositions of rank up to $O(n^2)$, with a leading coefficient improved over existing results.

Let $\phi$ and $\phi_0$ be the rational maps in the SAGBI homotopy setting, with degrees $\deg(\phi)$ and $\deg(\phi_0)$ respectively. Suppose that $\phi$ can be represented by a h…

Let ${\mathcal{O}}$ be a non-simplicial MaxDeg planar order ideal, and let $\mathbb{B}_{\mathcal{O}}$ denote its border basis scheme. A $Z$-separating re-embedding means a re-embed…

Let $X$ be the curve considered in the case $p=e_2$, and let $\delta(X,k)$ denote the delta invariant associated with rank-$k$ quadratic forms. Delta-gap conjecture. … The argument…

Let $A$ be a general positive $3 \times 3$ matrix, and let $\operatorname{Sink}(A)$ denote its Sinkhorn limit. An entry is said to have degree over a field if it is algebraic over…

Consider the polynomial systems produced from the rotated version of the DruWo2017 semidefinite-programming instance, whose sizes prevent direct hand analysis. Macaulay2 radical-ge…

Let a polynomial system arise from a weakly feasible semidefinite program, and let its radical ideal be computed using Macaulay2. Macaulay2 minimal-generator conjecture. In the ins…

Rojas's gcd conjecture. One should be able to pick out the isolated roots simply by computing the gcd of the resultants arising from two different generic perturbations.

Strang's conjecture. For a generic triangulation $\Delta$, the dimension of the spline space is precisely

Sturmfels–Uhler conjecture. The ideal $I_n$ is the radical ideal defining $X_{L_{C_n}}$.

Sa conjecture. The set of polynomials over finite fields that do not satisfy any of the conditions in Sa is empty.

First Betti number conjecture. In some cases, the first Betti number of $\mathrm{Gobs}(F)$ represents the phase of the monomial ideal generated by $\operatorname{LM}(F)$.

Generator-growth conjecture. The number of minimal generating elements of the ideal will increase dramatically as the dimension increases, thereby making Buchberger's algorithm com…

Let $\mathbb{K}$ be the coefficient field, let $\bold{x}$ denote the tuple of auxiliary variables, and let $p\in\mathbb{K}[\bold{x}][y]$ be a primitive polynomial. Let…