16 problems

Let $k$ be an algebraically closed field, let $V$ be a variety over $k$, and let $(V,\phi)$ be a $\sigma$-variety over $k$, with $\phi$ the self-map defining its difference structu…

Let $K$ be a difference field and let $R$ be a finitely difference generated difference algebra over $K$. A difference ideal in $R$ is radical well-mixed if it is both radical and…

Let $\mathbb{K}$ be a field of characteristic zero. Consider a discrete-time rational dynamical system … where $T,R_1,\ldots,R_k$ are rational functions over $\mathbb{K}$, and wher…

Let $k=(\mathbb{F}_q,\operatorname{id})$ and let $\bar{k}_n=(\overline{\mathbb{F}}_q,\varphi_n)$, where $\varphi_n(\alpha)=\alpha^{q^n}$. For a $k$-algebra $A$, define the numbers…

Let $k$ be a difference field and let $A$ be a $k$-algebra. The strong core of $A$, denoted $\pi_0^\sigma(A)$, is the union of all strongly $\operatorname{Diff}$-étale $k$-subalgeb…

Let $k=(\mathbb{F}_q,\operatorname{id})$ and let $\bar{k}_n=(\overline{\mathbb{F}}_q,\operatorname{Frob}_q^n)$ be the algebraic closure of the finite field equipped with the $n$-th…

Non-existence conjecture. Unless $K$ has cardinality less than $ba$ or is already $ba$-closed, $K$ does not have a $ba$-closure.

Let $k$ be an algebraically closed field, and let $(V,\phi)$ be a projective $\sigma$-variety over $k$. The action of $\phi$ on the numerical equivalence classes of divisors on $V$…

Pellarin's expansion conjecture. For all $n\in\mathbb{N}^*$ and $\Sigma\subset\mathbb{N}^*$ satisfying $|\Sigma|\equiv n\pmod{q-1}$, there is a unique expansion

Let $R$ be a difference polynomial ring, and let a binomial difference ideal be a difference ideal generated by binomials. Its radical well-mixed closure is the smallest radical we…

Dichotomy conjecture. Either $f$ or $\exp_C(f)$ is regularly $\mu$-transcendental over $K_s$.

Regular-independence conjecture. Every finite subset of this set is regularly $\mu$-independent over $K_s$.

Operator-theoretic generalization of Denis–Schanuel's conjecture. The asserted regular $\mu$-independence holds.

Let $k$ be a … -generated $k$- … pi0^(R)=pi0^(R|k), … -étale $k$- … -generated $k$- … -étale $k$- … -generated over $k$, or finite-dimensional as a $k$-vector space; the supplied t…

Let $k$ be a $\sigma$-field and $R$ a finitely $\sigma$-generated $k$-$\sigma$-algebra. A prime … of $R$ is minimal if every prime … with $\subset$ satisfies $=$. Minimal-prime fin…

Let $\mathcal{I}_{{\mathbf{u}}}$ be the elimination ideal associated with a Laurent transformally essential system, and let ${\mathbf{R}}$ be the sparse difference resultant of…