Related papers: Almost every real quadratic polynomial has a poly-…
We consider complex polynomials $f(z) = z^\ell+c_1$ for $\ell \in 2\N$ and $c_1 \in \R$, and find some combinatorial types and values of $\ell$ such that there is no invariant probability measure equivalent to conformal measure on the Julia…
We prove a version of the classical $\lambda$-lemma for holomorphic families of Riemann surfaces. We then use it to show that critical loci for complex H\'{e}non maps that are small perturbations of quadratic polynomials with Cantor Julia…
The purpose of this note is give some evidence in support of conjectures of Poonen, and Morton and Silverman, on the periods of rational numbers under the iteration of quadratic polynomials. In particular, Poonen conjectured that there are…
We consider quantum computational models defined via a Lie-algebraic theory. In these models, specified initial states are acted on by Lie-algebraic quantum gates and the expectation values of Lie algebra elements are measured at the end.…
We study polyharmonic (k-harmonic) maps between Riemannian manifolds with finite j-energies (j=1, cdots, 2k-2). We show if the domain is complete and the target is the Euclidean space, then such a map is harmonic.
We prove several new rigidity results for polynomial automorphisms of $\mathbb C^2$ with positive entropy. A first result is that a complex slice of the (forward or backward) Julia set is never a smooth, or even rectifiable, curve. We also…
A field $k$ is called large if every irreducible $k$-curve with a $k$-rational smooth point has infinitely many $k$-points. Let $k$ be a perfect large field and let $f \in k[x]$. Consider the evaluation map $f_k: k \to k$. Assume that $f_k$…
It was known that the Sierpi\'{n}ski carpets can appear as the Julia sets in the families of some rational maps. In this article we present a criterion that guarantees the existence of the carpet Julia sets in some rational maps having…
A period is a complex number arising as the integral of a rational function with algebraic number coefficients over a rationally-defined region. Although periods are typically transcendental numbers, there is a conjectural Galois theory of…
We construct a polynomial-time algorithm which given a graph $\Gamma$ finds the full set of non-equivalent Cayley representations of $\Gamma$ over the group $D\cong C_p\times C_{p^k}$, where $p\in\{2,3\}$ and $k\geq 1$. This result implies…
Let $f$ be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve $C$. We conjecture that this happens if and only if $f$ admits a…
Ehrhart's famous theorem states that the number of integral points in a rational polytope is a quasi-polynomial in the integral dilation factor. We study the case of rational dilation factors and it turns out that the number of integral…
The polynomial-time computability of the permanent over fields of characteristic 3 for k-semi-unitary matrices (i.e. square matrices such that the differences of their Gram matrices and the corresponding identity matrices are of rank k) in…
Lorentzian polynomials are a fascinating class of real polynomials with many applications. Their definition is specific to the nonnegative orthant. Following recent work, we examine Lorentzian polynomials on proper convex cones. For a…
We describe the statistical properties of the dynamics of the quadratic polynomials P_a(z):=e^{2\pi a i} z+z^2 on the complex plane, with a of high return times. In particular, we show that these maps are uniquely ergodic on their measure…
Necessary and sufficient conditions are obtained under which the numerator of the partial derivative of a rational function holomorphic in open upper poly-halfplane is the sum of squares of polynomials.
We are concerned with the behavior of the polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ with finite fibres and satisfying the condition that all of the curves $aP+bQ=0$, $(a:b)\in \mathbb{P}^1$, are irreducible rational curves. The obtained…
We give equations for 13 genus-2 curves over $\overline{\mathbb{Q}}$, with models over $\mathbb{Q}$, whose unpolarized Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order. If the…
A non-zero constant Jacobian polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ is invertible if $P$ and $Q$ are rational polynomials.
The recent work by Marcus, Spielman and Srivastava proves the existence of bipartite Ramanujan (multi)graphs of all degrees and all sizes. However, that paper did not provide a polynomial time algorithm to actually compute such graphs.…