Mathematics Problem Archive

{\rm (V. Retakh) Establish complete integrability of the noncommutative version of the leapfrog map. Define noncommutative versions of the pentagram m...

Consider the action of $ Aut (V,J)$ on $V$. Describe the partition of $V$ into $ Aut (V,J)$-orbits. More generally, describe the action of $ Aut (V,J)...

Find necessary and sufficient conditions for the bi-Lagrangian Grassmannian $LG(V,J)$ to be a smooth algebraic variety. Describe the partition of $LG(...

Do bi-integrable systems exist for each algebraic type?...

Are there any restrictions on the algebraic type of the pencils $\mathcal{A}_{x+\lambda a}$? Which algebraic types can be realised by means of an appr...

Study examples of ``quadratic $+$ linear'' Poisson pencils. Compute their algebraic types and construct complete families of polynomials in bi-involut...

Is it true that for any quadratic Poisson bracket (defined on a vector space), there exists a polynomial integrable system?...

Describe closed manifolds $M$ which admit Nijenhuis operators $L(x)$ that are algebraically regular at each point $x\in M$....

Let us fix a certain algebraic type of a linear operator, i.e., its Segre characteristic (see above). Does there exist a Nijenhuis operator $L$ in $\m...

Describe all the functions $f(x,y)$ of two variables defined in a neighbourhood of $(0,0)\in\mathbb{R}^2$ such that ; $f_y(0,0)\not\equiv 0$; ; $f_y(0...

Consider a smooth map $\Phi=(\sigma_1,\dots,\sigma_n): U(0) \to \mathbb{R}^n$, where $U(0)$ is a neighbourhood of the origin $0\in \mathbb{R}^n$. We a...

Describe/classify the collections of algebraically independent homogeneous polynomials $\sigma_1, \dots, \sigma_n$, $\deg \sigma_k = k$, in $n$ variab...

Describe the structure of singularities of $\Phi=(\sigma_1,\dots,\sigma_n): M \to \mathbb{R}^n$ in terms of the singular points of the recursion opera...

Which foliations with affine leaves can be described as the image of the moment map?...

Consider a Poisson manifold $M$ of even dimension such that it is symplectic on a dense set $U\subset M$. Assume that $M$ is endowed with a toric acti...

[S. V\ u Ng{\d o}c] Given a set of semiclassical operators that verify conditions [source label: item:self-adjoint ] and [source label: item:commute] ...

[S. V\ u Ng{\d o}c] Define (and detect) the quantum Chern class....

[\'{A}. Pelayo] Can one make progress in counting the number of fixed points by studying the spectrum of the quantisation of $\mu \colon M \to S^1$?...

What are necessary and/or sufficient conditions on a metric $g$ such that every polynomial integral of its geodesic flow is quantisable?...

[Miranda-Presas-Solha ] Modify this scheme to get finite dimensional representation spaces for focus-focus and hyperbolic singularities that still cap...

[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 $p \bullet q$....

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

How big can the set of Valiron deficiencies be for functions in the plane? It is known that $$ N(r,a)=T(r,f)+O\big(T(r,f)^{\frac{1}{2}+\varepsilon}\bi...

Let $f(z)$ be an entire function of finite order $\rho$, and let $n_1(r,a)$ denote the number of simple zeros of the equation $f(z)=a$. If \[n_1(r,a)=...

Under what conditions can $\sum\delta(a,f)$ be nearly $2$ for an entire function of finite order $\rho$? Pfluger proved that if $\sum\delta(a,f)=2$, t...

Arakelyan has proved that, given $\rho>\frac{1}{2}$ and a countable set $E$, there exists an entire function $f(z)$ of order $\rho$, for which all the...

If $f(z)$ is an entire function of finite order $\rho$ which is not an integer, it is known that (see Pfluger and Hayman ), \[\sum \delta(a,f)\leq 2-K...

Following the notation in Problem 1.7, if $f(z)$ is meromorphic in the plane of order $\rho$, it is conjectured by Pfluger , that for $a\neq b$ \[\lim...

If $f(z)$ is a meromorphic function of finite order with more than two deficient values, is it true that if $\sigma>1$, then \[\limsup_{r\to\infty}\fr...

If $f(z)$ is a meromorphic function of finite order with at least one finite deficient value, does the conclusion of Problem 1.10 hold?...

Edrei, Fuchs and Hellerstein ask if $f(z)$ is an entire function of infinite order with real zeros, is $\delta(0,f)>0$? More generally, is $\delta(0,f...

If $f(z)$ is an entire function of finite order $\rho$ and lower order $\lambda$ with real zeros, find the best possible bound $B=B(\rho,\lambda)$ suc...

For any function $f(z)$ in the plane, let $n(r)=\sup_a n(r,a)$ be the maximum number of roots of the equation $f(z)=a$ in $|z|<r$, and \[A(r) = \frac{...

If $f(z)$ is non-constant in the plane, it is known (see Hayman ) that \[ \alpha_f=\limsup_{r\to\infty}\frac{T(r,f)}{T(r,f')}\geq \begin{cases} \frac{...

The defect relation ([source label: 1.2]) is a consequence of the inequality (see Hayman ), which is called the ``second fundamental theorem'', $$ \su...

Under what circumstances does $f(z_0+z)$ have the same deficiencies as $f(z)$? It was shown by Dugu{\'e} that this need not be the case for meromorphi...