Related papers: Variations on the Arkhipov-Kara\v{c}uba Type Count…

Let $p$ be an odd prime and $d = p^{\tau}(p-1)$. In the spirit of Aritn's conjecture, consider the system of two diagonal forms of degree $d$ in $s$ variables given by \begin{equation*}\begin{split} a_1x_1^d + \cdots + a_sx_s^d = 0\\…

It is established that for every pair of additive forms $f=\sum_{i=1}^s a_i x_i^k, g=\sum_{i=1}^s b_i x_i^k$ of degree $k$ in $s>2k^2$ variables the equations $f=g=0$ have a non-trivial $p$-adic solution for all odd primes $p$.

Let $\mathcal{F}=\{f_1,\ldots,f_R\}$ be a family of forms of odd degrees at most $d$ in $s$ variables. We study the solutions to the system $f_1(\mathbf{x})=\ldots=f_R(\mathbf{x})=0$ of the form $x_i=y_ip_i$ with $|y_i|\leq Y_\mathcal{F}$…

In 1927, E. Artin conjectured that all non-square integers $a\neq -1$ are a primitive root of $\mathbb{F}_p$ for infinitely many primes $p$. In 1967, Hooley showed that this conjecture follows from the Generalized Riemann Hypothesis (GRH).…

Kurepa's conjecture states that there is no odd prime $p$ that divides $!p=0!+1!+\cdots+(p-1)!$. We search for a counterexample to this conjecture for all $p<2^{34}$. We introduce new optimization techniques and perform the computation…

We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p+1. An extension of…

We establish an effective Bertini-type theorem for hypersurfaces $X_f \colon f = 0$ defined over a finite field $k$ for which $f$ has no linear factors over the algebraic closure $\overline{k}$. Given a line $L$ defined over $k$ and a…

For each prime number $p$ and each integer $g \geqslant 5$, we construct infinitely many abelian varieties of dimension $g$ over $\overline{\mathbb{F}}_p$ satisfying the standard conjecture of Hodge type. The main tool is a recent theorem…

We show that under the assumption of Artin's Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over $\bar F_p(x)$ with arbitrary high rank and constant j-invariant. For odd primes p, this result follows from a…

We prove that an additive form of degree $d=2m$, $m$ odd, $m\ge3$, over the unramified quadratic extension $\mathbb{Q}_2(\sqrt{5})$ has a nontrivial zero if the number of variables $s$ satisifies $s \ge 4d+1$. If $3 \nmid d$, then there…

A minimal counterexample to the Erd\H{o}s-Gy\'arf\'as conjecture is a graph of minimum possible order and size with minimum degree at least 3 that contains no cycle whose length is a power of 2. Markstr\"om observed that any such graph must…

In the moduli space of degree d polynomials, the special subvarieties are those cut out by critical orbit relations, and then the special points are the post-critically finite polynomials. It was conjectured that in the moduli space of…

Let ${\cal P}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^+ := \{z \in \mathbb{C}: |z| \leq 1, \, \, \Im(z) \geq 0\}$$ be the closed upper half-disk of the complex plane. For…

A famous conjecture of Artin asserts that any integer $a$ that is neither $-1$ nor a square should be a primitive root (mod $p$) for a positive proportion of primes $p$. Moreover, using a heuristic argument, Artin guessed an explicit…

Let $p$ be a prime number with $p\equiv 1\mod 4$, let $\omega=\frac{1+\sqrt{p}}{2}$, let $\varepsilon>1$ be the fundamental unit of $\mathbb{Z}[\omega]$ and let $x$ and $y$ be the unique nonnegative integers with $\varepsilon=x+y\omega$.…

Ahmadi-Shparlinski conjectured that every ordinary, geometrically simple Jacobian over a finite field has maximal angle rank. Using the L-Functions and Modular Forms Database, we provide two counterexamples to this conjecture in dimension…

We obtain new bounds of exponential sums modulo a prime $p$ with binomials $ax^k + bx^n$. In particular, for $k=1$, we improve the bound of Karatsuba (1967) from $O(n^{1/4} p^{3/4})$ to $O\left(p^{3/4} + n^{1/3}p^{2/3}\right)$ for any $n$,…

In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (this case is known as…

Over a field of characteristic zero, it is clear that a polynomial of the form (X-a)^d has a non-trivial common factor with each of its d-1 first derivatives. The converse has been conjectured by Casas-Alvero. Up to now there have only been…

Let $f$ be a newform of even weight at least $4$, level $N$ and trivial character. Let $p\nmid N$ be an odd prime number that is ordinary for $f$ and let $K$ be an imaginary quadratic field satisfying a generalized Heegner hypothesis…