Mathematics Problem Archive

Are two independent random subsets of $\mathbb{Z}$ almost surely quasi-isometric as metric spaces? What is the answer for other Cayley graphs, such as...

Let $P$ be a finite $p$-group of order $n$ and let $w$ be any word. Is the probability that $w$ is satisfied in $P$ at least $1/n$?...

Let $M$ be the set of measurable real functions, and define $H_f(x,y)=(x,y+f(x))$ and $V_f(x,y)=(x+f(y),y)$. What group is generated by $\{H_f:f\in M\...

Determine the asymptotic length of the shortest nontrivial word that is a law in every group of order $2^n$....

Given a triple of positive integers $(k,d,n)$, calculate $\operatorname{rk}_k^\circ(kd,n).$...

The $k$-rank of a general form of degree $kd$ in $n$ variables is given by $$ \operatorname{rk}_k^\circ(kd,n)=\begin{cases} \min \left\{s\ge 1 | s\bin...

Given a triple of positive integers $(k,d,n)$, calculate $\operatorname{rk}_k^{\max}(kd,n).$...

For any positive integers $k,d$, the maximal $k$-rank $\operatorname{rk}^{\max}_k(kd,2)$ of binary forms equals $k$. Additionally, in the above notati...

Given $k \geq 3$ and a monomial $m$ of degree $kd$, determine the monomial $k$-rank $\operatorname{rk}_k(m)$....

Given $k \geq 3$ and a monomial $x^a y^b$ of degree $a+b=kd$, it is known that $\operatorname{rk}_k(x^{a}y^{b}) \leq \max(s,t)+1$, where $s$ and $t$ a...

Calculate the degree of $\widetilde{W}_{k,d}$ for perfect pairs $(k,d)$....

[Fr\"oberg's Conjecture, 1985] Let $f_1, \dots, f_r$ be generic forms of degrees $d_1, \dots, d_r$, respectively. Then the Hilbert series of the quoti...

[Fr\"oberg-Iarrobino Conjecture] Given generic linear forms $\ell_1, \ldots, \ell_r$ and a positive integer $d$, let $I$ be the power ideal generated ...

For $\mu \neq (d)$, does a generic $\mu$-power ideal have the same Hilbert function as in [source label: eq:RALF]?...

[] For generic forms $g_1,\ldots,g_r$ of degree $d>1$, the ideal $(g_1^k,\ldots,g_r^k)$ has the same Hilbert series as the one generated by $r$ generi...

It has been conjectured that each complete intersection $R=S/(f_1,\ldots,f_n)$ satisfies the WLP and also the SLP, see . Does the same hold for $R=S/(...

When are the WLP and the SLP true for $T_{n,d,k}$?...

For $R= S/(f_1,\ldots,f_r)$, where $f_1,\ldots,f_r$ are generic forms, does $R$ satisfy the $\mu$-Lefschetz property for all partitions $\mu$?...

For the ideal $I$ of $s$ general points in $\mathbb P^{n-1}$, what is the difference between the Hilbert series of the $m$-th symbolic power and the $...

$\mathcal{S}$ is cyclotomic if and only if $k[\mathcal{S}]$ is a complete intersection....

Are $B_{n,m}$ and $B'_{n,m}$ finite for any pair $(n,m)$ with even $n$?...

For any given pair $(n,m)$ with even $n$, $B'_{n,m}=\left(\frac{n}{2}\right)^{m-1}$....

Determine $\lim_{n\to\infty}\frac{B_{n,3}}{n^2}$....

For $n=1$ and given $p$, what are the (lengths of the) possible periods of $\phi$?...

For $n = 1$ and given $p$, find the minimal positive integer $i$ such that $\psi^i$ is the identity map on the space of polynomials of degree at most ...

Let $f$ be a form of odd degree $d$ in $E$. Is it true that $({\rm Ann}(f))_i = (f)_i$, for $i < (n-d)/2$?...

Let $f$ and $g$ be generic quadratic forms in $E$ and let $\ell_1$ and $\ell_2$ be two generic linear forms in $S$. Then the Hilbert series of $E/(f,g...

Determine for which orders Williamson matrices exist; such matrices give a construction of Hadamard matrices....

For each order $n$, determine the largest possible absolute determinant of an $n\times n$ matrix whose entries are all $1$ or $-1$....

Let $m,p\ge 2$, put $d=m+p-1$, and let $F(x)=(1,x,\ldots,x^d)$ be the rational normal curve. For $j=1,\ldots,mp$, let $X_j$ be the $p$-plane spanned b...

Find an integer cosine rule for integer triangles in integer trigonometry....

{\bf(IKEA problem.)} Classify all $n$-tuples of LLS-sequences for the angles that form integer $n$-gons....

Classify all combinatorial possible types of faces....

Classify all empty simplices of dimension $n$ up to lattice congruence....

Which $n$-gons are realizable as faces of an $m$-dimensional continued fraction? Here are two essentially geometrically different subcases: ; {\bf Fac...

Describe all finite two-dimensional sails (and the corresponding continued fractions)....

{\bf (Multidimensional IKEA problem.)} Describe the collections of the sails of the cones for all polytopes of a given combinatorial type....

{\bf (V. Arnold.)} Does there exist an algorithm to decide whether a given type of fundamental domain is realizable by a periodic continued fraction?...

{\bf (V. Arnold.)} Torus decompositions of integer noncongruent Klein sails are distinct....

{\bf (V. Arnold.)} Describe all torus decompositions that are realized by periodic two-dimensional continued fractions....

{\bf (V. Arnold.)} Classify continued fractions that correspond to the same cubic extension of the field of rational numbers....

Prove the existence of a cone for a single non-periodic combinatorial structure ($n\ge 3$)....

Find frequencies on $n$-dimensional continued fractions with the highest relative frequencies....

For every positive integer constant $C$ there exist only finitely many pairwise integer non-congruent faces with frequencies exceeding $C$....

Is that true that sum of all relative frequencies for all possible faces is finite for higher dimensions $(n\ge 3)$?...

In case of positive answer to the above question find the generalization of the Gauss map and compare the corresponding frequencies of faces with the ...

Find a natural generalization of the Farey tessellation to higher-dimensional hyperbolic geometry....

{\bf (Jacobi's last theorem.)} Let $K$ be a totally real cubic number field. Consider arbitrary elements $y$ and $z$ of $K$ such that $0<y,z<1$ (here ...

Generalize continued fractions to describe 3-bridge knots....

Prove the existence of energy minimizers for elastostatics for quasiconvex stored-energy functions $W$ satisfying $W(A)\to\infty$ as $\det A\to0^+$ ....