19 problems

Let a finite group model be a lattice-walk model with finite orbit group, and let $b$ be an admissible integer starting-coordinate parameter. Consider three-quadrant walks starting…

The parametric algebraic-form conjecture.

Let $f$ be an algebraic function analytic at $\infty$, and let $[n/n]$ denote its diagonal Padé approximants. A spurious pole is a pole of a Padé approximant in the sense defined i…

Let $p=P'$ and $q=Q'$ be polynomials, and let the polynomial moments (1.2) be the moments associated with $P$ and $q$. The Polynomial Composition condition (PCC) is that there exis…

Let $\frak S$ be a set of steps defining a slit-plane model, and let its generating function be the generating function that counts walks avoiding the forbidden half-line. A walk e…

Let $\frak S$ be a set of steps defining walks on the slit plane, and let $S(x,y;t)$ be the corresponding generating function. Say that the forbidden half-line is crossable without…

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

Consider a quarter-plane lattice-walk problem and its orbit sum. Zero-orbit-sum algebraicity conjecture. If the orbit sum is $0$, then the solution is algebraic. Zero orbit sum is…

Let $(c_n)_{n\geq 0}$ be the sequence discussed in the paper, and for $(u,v)\in\mathbb{Q}^2$ define $\widetilde c_n=w^n(u)_n(v)_n c_n$, where $(u)_n$ and $(v)_n$ are Pochhammer sym…

Let $y(x)=\sum_{k\geq 0}a_kx^k\in\mathbb{Q}[[x]]$ be D-finite. Assume that the sequence $(a_k)_{k\geq 0}$ has at most geometric growth, that there exists $N\in\mathbb{N}$ such that…

Let $\overline{\mathbb{C}(x)}$ be an algebraic closure of $\mathbb{C}(x)$, equipped with the derivation $d/dx$. An algebraic function is stable if its iterated formal integrals eve…

Let $B_1$ and $M_2(p)$ be as in the preceding construction, with $p_2$ the coefficient of $F^2$ in the cubic factor $P_3(F)$. Coefficient-field subextension conjecture. The field…

Let $B_1$ be the series under consideration and let $\mathbb{Q}(t,B_1)$ be the field extension generated by it over $\mathbb{Q}(t)$. Degree-four subfield conjecture. The extension…

Let $G(t;a,b)$ denote the generating function for the corresponding model, with parameters $t$, $a$, and $b$. Models 19 and 20 are the Kreweras and reverse Kreweras walks, respecti…

Let $k$ be a positive integer, and let a bounded body in ${\mathbb R}^{2k+1}$ have the property that the volume cut off by an affine hyperplane depends algebraically on the coordin…

Let $f(z,t)$ and $f^{\not\ominus}(z,t)$ be the bivariate generating functions under consideration, and let $\overline{f(z,t)}$ and $\overline{f^{\not\ominus}(z,t)}$ be the correspo…

Balanced algebraic-function conjecture. Every such $P(\mathcal C,z)$ admits a positive motherbody measure.

Non-algebraicity conjecture. The generating function $D(z)$ is not algebraic.

Let $P(x,y)=y^m+a_1y^{m_1}+\cdots+a_ny^{m_n}+x$ define an algebraic function, and let … be its optimal annihilating operator. Denote by $\mathfrak{D}(a_1,\ldots,a_n,x)$ the discrim…