36 problems

Let $N\geq 1$ be a natural number, and let $P_{2n+2l}$ be a sequence of integer polynomials defined in the source's equation (with coefficients bounded in modulus by $N$). Its limi…

Margulis's conjecture. There is a function $\ell:\mathbb N\to\mathbb R^{>0}$ such that $m(p)\geq1+\ell\bigl(d(p)\bigr)$ for every non-cyclotomic monic polynomial…

For odd integers $n$, let $\alpha_n$ be the Gaussian-period algebraic integer and let $C_{\kappa(n)}$ be the associated cyclovariety, with Mahler measures denoted by…

Let $\alpha_n$ be the Gaussian-period algebraic integer defined in the paper, and let $\mathrm{m}(\alpha_n)$ denote its Mahler measure. For a property of odd integers $n$, say that…

Limiting alternative-measure conjecture. The sequence has a limit, and

Let $c(P)$ be the alternative measure of a polynomial, and let $k\geq 2$. Consider the integer-exponent family $1+x+x^{n_2}+x^{n_3}+\cdots+x^{n_k}$ and take the iterated limit with…

Alternative-measure conjecture.

Adelic hypersurface Lehmer conjecture. There exists a positive constant $c(n)$, depending only on $n$, such that

Let $P(x)\in\mathbb{Z}[x]$ be a monic irreducible polynomial, and let $m_{\mathbb{H}}(P)$ denote its quaternionic Mahler measure. Quaternionic Lehmer conjecture. There is a constan…

Let $m(P)$ denote the Mahler measure of a Laurent polynomial $P$. Let $E$ be the elliptic curve … which has conductor $15$. Boyd and Rodriguez Villegas's conjecture. The Mahler mea…

Let $f\in\mathbb{Z}[z]$ be a fixed monic polynomial of degree $d\geq 2$, and let $\mathrm{m}_f(P)$ denote the dynamical Mahler measure of a single-variable polynomial…

Let $\mathcal{L}$ be a $\mathbb{Z}^2$-periodic alternating link, with toroidally alternating quotient link $L$. Let $P(z,w)$ be the characteristic polynomial of the toroidal dimer…

Canonical-weight criterion conjecture. Up to $\operatorname{SL}(2,\mathbb{Z})$, the condition

Hauptmodul and monodromy-group conjecture. Then $k^n$ is a Hauptmodul for $\Gamma^b$,

Let ${\cal A}$ be a cluster algebra, or let $\varphi$ be a map with the Laurent property obtained from a cluster algebra with periodicity in the sense of the cited constructions. L…

Let $P(z,w)=k-p(z,w)$ be a Newton polynomial with fixed $p(z,w)$. Let $m_l(P)$ and $m_g(P)$ denote the liquid and gas contributions to the Mahler measure, and let $\mathcal{A}(P)$…

Consider Seiberg-dual dimer models with canonical edge weights, and let $P_1(z,w)$ and $P_2(z,w)$ be their Newton polynomials. Let $n_i$ be the number of edges, equivalently multip…

Let $P(\boldsymbol{z})=k-p(\boldsymbol{z})$ be a Laurent polynomial in $\boldsymbol{z}\in\mathbb{C}^n$, and let $\mathcal{A}(P)$ be its amoeba. Write $V_\textup{bdd}$ for the volum…

Let $P(z,w)=k-p(z,w)$ be a Newton polynomial undergoing the Mahler flow, with Mahler measure $m(P)$ and flow parameter $k>0$. Mahler-flow monotonicity conjecture. The Mahler measur…

Let $\theta_n^{-1}$ denote the relevant Perron-number sequence, and let a Salem number be a real algebraic integer greater than $1$ whose conjugates satisfy the usual Salem-number…

Let $M_{\inf}$ be the limit infimum of the Mahler measures ${\rm M}(\alpha)$ as $\lvert\alpha\rvert\to1^+$, and let $\Lambda$ be the limiting constant introduced for the relevant f…

Let $M$ denote Mahler's measure, and write $M^{(n)}$ for its $n$-fold iterate. For an algebraic unit $\alpha$, consider the sequence of real numbers…

Let $d\geq 1$, let $P\in \overline{\mathbb{Q}}[X_1^{\pm 1},\ldots,X_d^{\pm 1}]\setminus\{0\}$, and let $m(P)$ denote the logarithmic Mahler measure of $P$. For…

Let $\mathcal{L}$ be any biperiodic alternating link, with toroidally alternating $\Lambda$-quotient link $L$. Let $p(z,w)$ be the characteristic polynomial of the toroidal dimer m…

Let $\mathcal L$ be a biperiodic alternating link with alternating quotient link $L$, and let $K_n$ be alternating hyperbolic links satisfying…