13 problems

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

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 $\mathbb{K}$ 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…

Let $\mathcal{P}$ be the polynomial algebra under consideration, and let $\mathcal{G}\subset\mathcal{P}\setminus\{0\}$ be a finite set whose polynomials involve monomials of length…

Let $X\subset\mathbb{R}^n$ and let $\mathcal{R}_n$ denote the polynomial ring in $n$ variables. For a set $G\subset\mathcal{R}_n$, write $G^{\mathrm{deg}(g)-1}$ for the relevant de…

Initial minors conjecture. For a matrix $\mathcal{A}$ coming from the above approach to solve ECDLP, there exists a set of initial minors whose cardinality is bounded by a sub-expo…

Consistency conjecture. If

Let $F({\mathbf z})\in\mathbb{Z}(z_1,\dots,z_n)$ be a rational function, and let (A5) denote that the Jacobian matrix of the relevant system of equations (GenSys1)–(GenSys5) has fu…

Asymptotic certificate-degree conjecture. For every field $\mathbb K$ and every positive integer $m$, there exists a constant $k_0$ with the following property: for each $k>k_0$ an…

Moreno-Socias's conjecture. There is a non-empty Zariski-open subset $U\subseteq A_{d_1}\times\dots\times A_{d_s}$ such that every $(f_1,\dots,f_s)\in U$ generates an ideal…

Let $n\ne2$ and let $a_i,b_i\in\mathbb{Q}$ satisfy $0<a_i,b_i<1$ for $i=1,2,\ldots,n$. Let $F(a\mid z)$ be the generalized hypergeometric solution and let $G(a\mid z)$ be its first…

Let $d$ denote the current degree in the F5 algorithm, and let $d_{\textrm{FR}}$ be the degree at which all Gröbner-basis-critical (GB-critical) pairs have been rejected by the F5…

Let $f$ be the polynomial considered in the source, let $Kf$ denote its associated polynomial vocabulary description, let $\Omega_1(f)$ be the number of irreducible factors of $Kf$…