Mathematics Problem Archive

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

[ 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, $$ \#_{r}P_{i}(x) \le \min\{{\deg{P_i(x)},k}\}. $$...

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...

If $f(z)$ is meromorphic of finite order $\rho$ and $\sum\delta(a,f)=2$, it is conjectured that $\rho=n/2$, where $n$ is an integer and $n\geq 2$, and...

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...

(Edrei's spread conjecture) If $f(z)$ is meromorphic in the plane and of lower order $\lambda$, and if $\delta=\delta(a,f)>0$, is it true that, for a ...

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{...

(Paley's conjecture) For any entire function $f(z)$ of finite order $\rho$ in the plane, we have \[1\leq\liminf_{r\to\infty}\frac{\log M(r,f)}{T(r,f)}...

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...

If $f$ is meromorphic in the plane, can $n(r,a)$ be compared in general with its average value \[A(r)=\frac{1}{\pi}\int\int_{|z|<r}\frac{|f'(z)|^2}{(1...

In the opposite direction to Problem 1.24, does there exist a meromorphic function such that for every pair of distinct values $a, b$, we have \[\lims...

The analogue of Problem 1.7 may be asked for meromorphic functions. The proposers conjecture that in this case \[\sum\delta(a,f)\leq\max\{\Lambda_1(\r...

Let $E$ be the set for which $m(r,a)\to\infty$ as $r\to\infty$. How large can $E$ be if: [(a)] ; $f$ is entire and of order $\frac{1}{2}$ mean type, ;...

Are there upper bounds of any kind on the set of asymptotic values of a meromorphic function of finite order? (D. Drasin and A. Weitsman)...

Can one establish an upper bound on the number of finite asymptotic values of a meromorphic function $f(z)$ in $\mathbb{C}$, taking into account both ...

Let the function $f$ be meromorphic in the plane, and not rational, and satisfy the condition $$ \frac{T(r,f)}{(\log r)^3}\to\infty,\hspace{1cm}\text{...

Let $f$ be meromorphic in $\mathbb{C}$, and let $f^{-1}$ denote any element of the inverse function that is analytic in a neighbourhood of a point $w$...

Let $f$ be a meromorphic function of finite order $\rho$. Does the condition \[N(r,1/f')+2N(r,f)-N(r,f')=o(T(r,f)),\hspace{1cm}\text{ as }r\to\infty,\...

Let $n_1(r,a,f)$ denote the number of simple zeros of $f(z)-a$ in $\{|z|\leq r\}$. Selberg has shown that if: [(a)] ; $f$ is a meromorphic function of...

Determine the upper and lower estimates for the growth of entire and meromorphic solutions of algebraic ordinary differential equations (AODE). (This ...

Let $F$ be a polynomial in two variables, and let $y$ be a meromorphic solution of the algebraic ordinary differential equation $F(y^{(n)},y)=0$. Is i...