Mathematics Problem Archive

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 $-1$....

[ Maxwell, seems bad, no tools] For any system of $N$ isolated fixed point charges in $\mathbb{R}^3$, the number of points of equilibrium (assumed fin...

[folklore, very irritating] For any set of charges of the same sign in $\mathbb{R}^n$, the set of its points of equilibrium is finite....

[A. Gabrielov, D. Novikov, B. Sh., seems good, but no progress] Let $(x_1,y_1),(x_2,y_2),\dots, (x_N,y_N)$ be a collection of points in $\mathbb{R}^2$...

[B. Sh., looks bad, but very important] Does there exist an upper bound for the number of real roots valid for all non-trivial solutions of all equati...

[D. Khavinson, I. Itenberg, B. Sh., apparently bad] Find the maximal possible number $\#(2k, l)$ of isolated zeros for real non-negative polynomials o...

[G. Ottaviani, B. Sh., seems good] Find the maximal possible number $\widetilde\#(2k, l)$ of isolated zeros for real non-negative polynomials of degre...

[G. Ottaviani, B. Sh., seems good] For any number of variables, $\widetilde\#(2k,l)=k^l$....

[S. Fisk, seems bad, see , p. 575] Given a pair of real polynomials $(p,q),$ give restrictions on the location of the roots of $p+iq$ in terms of the ...

[P. Br\"anden, I. Krasikov, B. Sh., hopefully good, see ] A difference operator $T(p(x))=a_0p(x)+a_1p(x-1)+\cdots+a_kp(x-k)$ with constant coefficient...

If $p$ and $q$ are real-rooted polynomials of degree at most $d$ and of mesh $\geq 1$, then so is $p \bullet q$....

[seems bad, but might be ugly] For a given sign pattern $\sigma,$ which admissible pairs $(pos,neg)$ are realizable by polynomials whose signs of coef...

[J. Forsg\aa rd, V. Kostov, B. Sh, hopefully good, see ] For an arbitrary sign pattern $\sigma$, the only type of pairs $(pos,neg)$ which can be non-r...

[J. Forsg\aa rd, B. Sh., seems good, see ] Let $f(z) = \sum_{k=0}^n a_k z^k$ be a polynomial with positive coefficients, and consider the related (wei...

[J. Forsg\aa rd, B. Sh., seems good, see ] Let $f(z) = \sum_{k=0}^n a_k z^k$ be a polynomial with positive coefficients. Consider the differences \[ \...

[J. Forsg\aa rd, B. Sh., seems good, see ]] Let $f(z) = \sum_{k=0}^n a_k z^k$ be a polynomial with positive coefficients. Consider the differences \[ ...

[V. Kostov, B. Sh., looks ugly, see ] What additional restrictions besides [source label: eq:1] exist on configurations $\mathcal A_{f}=\{x^{(i)}_{l}\...

[looks ugly] What symbolic sequences can occur for strictly real-rooted polynomials of degree $n$?...

[B. Sh., seems good] For any real polynomial $p(x)$ of degree $k$ with simple real zeros, $$ \#_{r}\left[(k-1)(p'(x))^2-kp(x)p''(x)\right] \le \#_{nr}...

[B. Sh] For any real polynomial $p(x)$ of even degree, $$ \#_{r}\left[(k-1)(p'(x))^2-kp(x)p''(x)\right] + \#_{r}p(x)>0, $$...

[B. Sh] For any degree $k$ polynomial $p(x)$ with real coefficients, $$ \#_{r}P_{i}(x) \le \min\{{\deg{P_i(x)},k}\}. $$...

Let $f(x_1,\ldots,x_n)\in\mathbb{C}[x_1,\ldots,x_n]$ satisfy $$M_m:=\int_0^1\cdots\int_0^1 f(x_1,\ldots,x_n)^m\,dx_1\cdots dx_n=0\qquad(m\geq1).$$ Mus...

Let $f(x)\in\mathbb{C}[x]$ have $k$ monomials. Does there exist $N(k)$ such that if $$M_n:=\int_0^1f(x)^n\,dx=0\qquad(1\leq n\leq N(k)),$$ then $f=0$?...

Find a reasonable, or sharp, upper bound in terms of $m_1,m_2$ for the maximum number of simple positive-coordinate solutions of a real polynomial sys...

Is $(2m_1-1)(2m_2-1)$ the maximum number of simple solutions of a real polynomial system $f_1(x,y)=f_2(x,y)=0$, where $m_i$ is the number of monomials...

For the classical-group pairs and generic local $L$-parameters in the Gan–Gross–Prasad setting, is there exactly one relevant representation $\pi$ in ...