Mathematics Problem Archive

The $d-1$ numbers $$ \alpha^\beta,\; \alpha^{\beta^2},\ldots, \alpha^{\beta^{d-1}} $$ are algebraically independent over $\mathbb{Q}$....

The $d$ numbers $$ \log\alpha,\; \alpha^\beta,\; \alpha^{\beta^2},\ldots, \alpha^{\beta^{d-1}} $$ are algebraically independent over $\mathbb{Q}$....

Let $\beta_1,\ldots,\beta_n$ be $p$-adic algebraic numbers in the domain of convergence of the $p$-adic exponential function $\exp_p$. Then the $n$ nu...

Let $\alpha$ be a non-zero algebraic number in the domain of convergence of the $p$-adic logarithm $\log_p$, and let $\beta$ be a $p$-adic cubic algeb...

Given an absolute constant $c$ and polynomials $P_1,\ldots,P_m$ with a total of $N$ coefficients and no common complex zeros, there is no program to f...

Let $\Sigma$ be a finite subset of $\mathbb{C}^n$ and $\varepsilon$ a positive number. There exists a positive number $r_0(\Sigma,\varepsilon)$ such t...

At least three of the four numbers $$ \pi,\; \Gamma(1/5),\; \Gamma(2/5), \; e^{\pi\sqrt 5} $$ are algebraically independent over $\mathbb{Q}$....

Let $\tau\in\mathbb{C}$ have positive imaginary part. Assume that $\tau$ is not quadratic. Set $q=e^{2i\pi\tau}$. Then at least $4$ of the $5$ numbers...

Let $\mathcal{E}_1,\ldots,\mathcal{E}_n$ be pairwise non isogeneous elliptic curves with modular invariants $j(\mathcal{E}_h)$. For $h=1,\ldots,n$, le...

Given an elliptic curve with Weierstrass equation $y^2=4x^3-g_2x-g_3$, a nonzero period $\omega$, the associated quasi-period $\eta$ of the zeta funct...

Let $q_1,\ldots,q_n$ be nonzero algebraic numbers in the unit open disc such that the $3n$ numbers $$ J(q_i), \; DJ(q_i),\; D^2J(q_i)\qquad (i=1,\ldot...

Let $q_1$ and $q_2$ be two nonzero algebraic numbers in the unit open disc. Suppose that there is an irreducible element $P\in\mathbb{Q}[X,Y]$ such th...

Is there such a bound depending polynomially on the degree and height of $P$?...

Are there entire transcendental functions $f(z)$ such that if $x$ is a Liouville number then so is $f(x)$?...

For each positive integer $n\ge 1$ there exists a positive number $c(n)$ having the following property. Let $\alpha_1,\ldots,\alpha_n$ be multiplicati...

For each positive integer $n\ge 1$ there exists a positive number $c(n)$ such that, if $\underline{\alpha}=(\alpha_1,\ldots,\alpha_n)$ is a $n$-tuple ...

For each integer $n\ge 1$ there exists a positive constant $c(n)$ such that, for any algebraic subvariety $V$ of $\mathbb{G}_m^n$ which is defined ove...

For $\theta\in(0,\pi)$, define $$ V_\theta=\{re^{it}\; ;\; r>0,\; |t|>\theta\}. $$ Compute $L(V_\theta)$ in terms of $\theta$....

There exists a positive constant $c$, depending only on $A$ and $\mathcal{L}$, such that for any $P\in A(\overline{\mathbb{Q}})$ which has infinite or...

Let $\underline{\theta}=(\theta_1,\ldots,\theta_m)$ be a $m$-tuple of complex numbers. Define $$ t=\operatorname{trdeg} \mathbb{Q}(\underline{\theta})...

Let $\theta\in\mathbb{C}^m$. There is a positive constant $c$, depending only on $\theta$ and $m$, with the following property. Let $k$ be an integer ...

There exist two positive absolute constants $c_1$ and $c_2$ with the following property. Let $\lambda_1,\ldots,\lambda_m$ be logarithms of algebraic n...

There exists a positive absolute constant $C$ with the following property. Let $\alpha_1,\ldots,\alpha_n$ be nonzero algebraic numbers and $\log\alpha...

Let $A$ be a simple abelian variety over $\mathbb{Q}$, $\exp_A:\mathbb{R}^g\rightarrow A(\mathbb{R})^0$ the exponential map of the Lie group $A(\mathb...

Let $m$, $n$, $k$ be positive integers and $a_{ij\kappa}$ rational integers ($1\le i\le n$, $1\le j\le m$, $1\le\kappa\le k$). For $\underline{x}=(x_1...

For any $\varepsilon>0$ there exists $S_0>0$ (depending on $\varepsilon$, $\gamma_1,\ldots,\gamma_m$ and $\mathcal{K}$) such that, for any $S\ge S_0$ ...

Let $n \ge 2$. Denote by $\mathrm{ZH}_n$ the set of real numbers $\xi$ with the following property: there exists $c_1(\xi) >0$ and $c_2(\xi)>0$ such t...

Let $(n_i)_{i\ge 0}$ be an increasing sequence of positive integers. Assume there is a prime number $p$ such that the power series $$ \sum_{i\ge 0}z^{...

Given an elliptic curve $E/\mathbb{F}_q$ and $P\in E(\mathbb{F}_q)$, characterize maps or isogenies $\psi:E\to E'$ for which $\psi(P)$ can be evaluate...

Find bilinear structures that are useful for cryptographic constructions....

Find orders $\mathcal{O}$ whose Picard groups contain $\mathbb{Z}/\ell$, admit compact element representations, and allow group composition in $O(\log...

Make the proposed realization of existing finite-field discrete-logarithm systems inside class groups explicit for practical systems....

Determine whether Weil descent compromises the security of the proposed class-group realization of finite-field discrete-logarithm systems....

Develop a fast algorithm to compute the Tate–Lichtenbaum pairing $T_n$....

Turn $H^2(G_K,K_s^*)$ into an explicitly computational group....

Explicitly describe the cocycle $c_u$, equivalently fast-compute invariants of local algebras split by the generalized-dihedral extensions specified i...

Explicitly construct global algebras or Brauer classes with prescribed local data, especially when the local splitting fields are dihedral....

Adapt Schoof's method to compute zeta functions of curves without unfavorable dependence on the genus....

Determine whether natural de Rham cup-product pairings can be used to improve zeta-function computations....

Remove the imaginary-hyperelliptic and $p\ne2$ restrictions from the complexity bound stated in the slides....

Test and analyze whether deleting extra points and fixing the lift $x\mapsto x^p$ improves Frobenius lifts for nondegenerate curves....

Develop the higher-dimensional analogue of the nondegenerate-curve Frobenius-lift method....

Find useful deformations of nondegenerate curves together with easy starting matrices for Frobenius computation....

Work out effective zeta-function computations for nondegenerate surfaces in toric threefolds....

Factor $N=p^rq^s$ with $r\approx s$ using fewer known bits of the factors than existing methods require....

Factor $N=pqr$ from fewer known bits of its prime factors....

Develop methods to factor an integer when the known bits of its factors are nonconsecutive....

Reduce the number of guesses required by lattice attacks for factoring with partially known bits....

Extract useful information from incorrect guesses in factoring attacks based on partially known bits....

Efficiently solve for, or characterize all relevant roots of, $x^2-1$ modulo a composite integer $N$ in the setting of the slides....