53 problems

Formal convergence conjecture. In the $x_i$ variables, the partition functions converge as formal power series expansions.

Positivity conjecture. All the coefficients $C_G$ of the elements $R_g\in\mathcal{W}$ are positive.

Let $\mathfrak g$ be a finite-dimensional Lie algebra, let $n$ be a positive integer, and let $x_1,\ldots,x_n\in\mathfrak g$. Define … For each $i$, let $\partial_{x_i}F_i$ be the…

Cubic-series form conjecture. The solution has the form

Algebraicity criterion. The generalized double Laurent series is algebraic over $\mathbb{F}_q(t,u)$ if and only if: (a) for some positive integers $a$ and $b$,…

Let $S(x)$ be the generating function of ternary square-free words, obtained in the limit $\ell\rightarrow\infty$ from the finite-length generating functions $S^{(\ell)}(x)$. Natur…

Let $S^{(\ell)}(x)$ be the generating function for ternary square-free words of length at most or equal to $\ell$, and let its poles be the zeros of its denominator polynomial. Pol…

Let $F$ be an analytic map defined in an open neighborhood of $0\in{\mathbb C}^n$, and let $a(z)$ denote the formal series associated with $F$ through the formal differential opera…

Let $r(x)$ be a monic polynomial of degree $N+1>1$ with distinct roots, and let $\lambda(x)$ and $\mu(x)$ be the unique polynomials satisfying … with $\deg\mu(x)\leq N-1$ and…

Divergence criterion. The map $F(z)$ is polynomial if and only if $G(z)$ is polynomial. The source presents this as an algebraic formulation equivalent to the Jacobian conjecture,…

Let $f(z)=\sum_{j\geq 0}\gamma_j z^j$ be a formal power series, let $f_n(z)=\sum_{j=0}^{n}\gamma_j z^j$ be its partial sums, and set $s_n=f_n(z)$. For $k,n\in\mathbb{N}_0$, define…

Samuel's conjecture. The ring $R[[t]]$ is a unique factorization domain.

Let $G$ be a finitely presented group, and let its conjugacy growth series be the formal power series whose coefficients are the values of the conjugacy growth function. Ciobanu–Ev…

Greedy 0-monoid conjecture. The formal power series $g(x)$ is a regular semi-upho function if and only if it admits an infinite greedy $0$-monoid series $\{M_k^0\}_{k\ge 1}$.

Greedy LCH-series conjecture. The propositions establishing that an infinite greedy LCH monoid series produces a regular upho function, and that the associated coefficient inequali…

Let $P=x^2y^2+(x^2+x+1)y+x^2$, and let $\theta$ be its Furstenberg series. For each $\alpha\geq 1$, consider the orbit of $\theta$ under the operator $\lambda_{0,0}$ modulo…

Catalan-polynomial structure conjecture. There exists a unique power series $P(x)$ with non-negative integer coefficients such that every $V_i(z)$ is the product of $C(z)$ and a po…

Let $\mathbb{F}_s$ be the algebraic closure of $\mathbb{F}_q$ in $K_v$, let $z$ be a uniformizer, and identify $K_v=\mathbb{F}_s((z))$ and $A_v=\mathbb{F}_s[[z]]$. Let…

Let $S$ be an additively reduced and additively atomic semidomain, and let $cmathcal{A}_{+}(S)$ denote the set of additive atoms of $S$ and $S^{\times}$ its multiplicative unit gro…

For each $k\geq 1$, let $A_k(x):=\sum_{n\geq 0}a_{k,n}x^n$ be the modified formal power series introduced from the arrival recurrence, and let $A_k'(0)$ denote its derivative at ze…

Let $G$ be a group with finite generating set $X$. Let $C_{\mathrm{Red}}(z)$ denote the generating series counting freely reduced words over $X$ that represent the identity in $G$.…

Let $k$ be the coefficient field, let $d_1,d_2,d_3$ be nonnegative integers, and write $x=(x_1,\ldots,x_{d_1})$, $y=(y_1,\ldots,y_{d_2})$, and $z=(z_1,\ldots,z_{d_3})$. For a tuple…

Let $p\geq 2$ and let … Suppose … is $p$-periodic, where $\varphi_j$ denotes the homogeneous degree-$j$ part. Zero-radius conjecture. If $\varphi\ne\varphi_1$, then the radius of c…

Formal identity conjecture. One has

The Valqui–Guccione–Guccione criterion. Then