8 problems

Let $R$ be a ring with an ideal $I$, let $n\geq 1$, and let $(R,I)\langle X_1,\ldots,X_n\rangle$ be the Tate algebra of formal power series whose coefficients converge to zero in t…

Let $\mathcal{L}(K)$ be the $K$-algebra generated by the lambda values, equipped with the lambda-degree induced by the lambda-type of composition arrays and taking values in the mo…

For $n\in\mathbb{N}^*$ and $\Sigma\in\mathfrak{S}$, let $\boldsymbol{C}_{n,\Sigma}$ be the corresponding space of composition arrays, and let $\mathcal{Z}_{n,\Sigma}$ and…

Modular analogue conjecture. For all $m\geq0$, if $|\Sigma|=s=m(q-1)+1$ and $q>m$, then

Harmonic-product conjecture. The only $\mathbb{F}_p$-algebraic relations in $\mathbb{I}$ are those coming from the harmonic product.

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

The correspondence conjecture. The correspondence $\zeta_A(\mathcal{C})\leftrightarrow\varphi_A(\mathcal{C})$ induces an isomorphism of $\mathbb{F}_p$-algebras

Carlitz–Schanuel conjecture over Tate algebras. The displayed transcendence-degree inequality holds.