19 problems

Let $f(x)$ be a globally bounded D-finite series, meaning a D-finite series with positive radius of convergence that can be reduced modulo almost all primes. A multivariate power s…

Diagonal-positivity conjecture. The series $\operatorname{diag}_F$ is nonnegative if and only if

Let $F(\mathbf{x})$ be a multivariate algebraic power series with integer coefficients, and let $\operatorname{Diag}(F)$ denote its diagonal. For each prime number $p$, reduce the…

The algebraicity–diagonal–global-boundedness conjecture. The following statements are equivalent:

For $p,q\in\mathbb{N}\setminus\{0\}$, define the Apéry-like power series … Let $\mu_{p,q}$ be the minimal number of variables needed to represent $f_{p,q}(z)$ as the diagonal of a…

Let $fin \mathbb{Q}[[z]]$ be a globally bounded and D-finite power series, and let $L^{\mathrm{min}}_f$ denote its minimal annihilating differential operator. Christol–André conjec…

Let $f$ be a diagonal power series, and let its Hadamard grade be the least number of diagonals whose Hadamard product equals $f$, when such a finite number exists. Lang–Rohrlich c…

Equal-logarithm global-boundedness conjecture. In this case, $T_0$ will necessarily be globally bounded and, assuming Christol's conjecture, will be the diagonal of a rational func…

Lower-logarithm non-global-boundedness conjecture. In the case $m<n$, the series $T_0$ cannot be globally bounded.

Highest-logarithm global-boundedness conjecture. The series $S_0$ will necessarily be globally bounded. Consequently, assuming Christol's conjecture, it will be the diagonal of a r…

Parity conjecture. The parity of the minimal number $N_v$ of variables determines the character of the differential Galois group: it is symplectic, contained in $Sp$, when $N_v$ is…

Minimal-variable conjecture. The diagonal of $R_1$ should coincide with the diagonal of a rational function $R$ depending on a minimal number $N_v$ of variables, with

Let $f(\boldsymbol t)\in\mathbb Z[[\boldsymbol t]]$ be an algebraic power series over $\mathbb Q(\boldsymbol t)$, and for each prime number $p$ let…

Let $G$ be a virtually abelian group, let $\mathcal S$ be a generating set, and write $\operatorname{Cog}_{\mathcal S}(t)$ for its cogrowth series. Virtually abelian cogrowth conje…

Diagonal conjecture. If the generating series of a primary pseudo-polynomial is the diagonal of a rational fraction, then $(a_n)_{n\geq 0}$ is a polynomial.

Let $C_n=\frac{1}{n+1}\binom{2n}{n}$ be the Catalan numbers, and let an $\mathbb{N}$-rational function mean a rational function with the relevant nonnegative-integer coefficient st…

A rank $n$-diagonal is a diagonal satisfying the rank-$n$ condition described in the surrounding theory. Diagonal-rank separation conjecture. For every natural number $n$ there is…

A linear differential operator is an operator annihilating a diagonal of a rational function, and an irreducible factor is an irreducible differential-operator factor of such an op…

Adjoint-homomorphism conjecture. Every irreducible factor of $L$ should be homomorphic to its adjoint, possibly after passing to an algebraic extension.