We discover two infinite families of integers $(P,Q,R,S)$ such that the polynomial $(((X^2-P)^2-Q)^2-R)^2-S^2$ factors into linear factors.

For continuous interval maps we formulate a conjecture on the shape of the cycles of maximum topological entropy of period $4k+2$ We also present numerical support for the conjecture. This numerical support is of two different kinds. For periods $6$, $10$, $14$, and $18$ we are able to compute the maximum-entropy cycles using nontrivial ad hoc numerical procedures and the known results of Jungreis, 1991. In fact, the conjecture we formulate is based on these results.

For periods $n=22$, $26$, and $30$ we compute the maximum-entropy cycle of a restricted subfamily of cycles denoted by $C_n^\ast$. The obtained results agree with the conjectured ones. The conjecture that we can restrict our attention to $C_n^\ast$ is motivated theoretically. On the other hand, it is worth noticing that the complexity of examining all cycles in $C^\ast_{22}$, $C^\ast_{26}$, and $C^\ast_{30}$ is much less than the complexity of computing the entropy of each cycle of period $18$ in order to determine those with maximal entropy, therefore making it a feasible problem.

We study the integer sequence $v_n$ of numbers of lines in hypersurfaces of degree $2n-3$ of $\P^n$, $n>1$. We prove a number of congruence properties of these numbers of several different types. Furthermore, the asymptotics of the $v_n$ are described (in an appendix by Don Zagier). Finally, an attempt is made at carrying out a similar analysis for numbers of rational plane curves.

Let $F$ be a real quadratic field with narrow class number one, and $f$ a Hilbert newform of weight $2$ and level $\mathfrak{n}$ with rational Fourier coefficients, where $\mathfrak{n}$ is an integral ideal of $F$. By the Eichler--Shimura construction, which is still a conjecture in many cases when $[F:\Q]>1$, there exists an elliptic curve $E_f$ over $F$ attached to $f$. In this paper, we develop an algorithm that computes the (candidate) elliptic curve $E_f$ under the assumption that the Eichler--Shimura conjecture is true. We give several illustrative examples that explain among other things how to compute modular elliptic curves with everywhere good reduction. Over real quadratic fields, such curves do not admit any parameterization by Shimura curves, and so the Eichler--Shimura construction is still conjectural in this case.

A conjecture that expresses the {$n$}th power of the cubic theta function $a(q)=\sum_j\sum_k q^{j^2+jk+k^2}$ in terms of Eisenstein series is formulated. It is an analogue of four conjectures of H. H. Chan and K. S. Chua for powers of $\varphi^2(q)=\sum_j\sum_k q^{j^2+k^2}$. With the help of a computer, the conjecture is shown to be true for $6\leq n \leq 100$. It is conjectured that the result continues to hold for $n>100$.

Let $\alpha$ be an algebraic integer all of whose conjugates lie in a sector $ | \operatorname{arg {\ z | \leq \theta$} with $ 0 \leq \theta <90^\circ$. Using the method of explicit auxiliary functions, we compute the greatest lower bound $v(\theta)$ of the absolute trace of $\alpha$, for $\theta$ belonging to seven subintervals of $[0,90^\circ)$. The polynomials involved in the auxiliary functions are found by Wu's algorithm.

We determine the minimal Mahler measure of a primitive irreducible noncyclotomic polynomial with integer coefficients and fixed degree $D$, for each even degree $D\leq54$. We also compute all primitive irreducible noncyclotomic polynomials with measure less than $1.3$ and degree at most $44$.

We describe a general method to compute weight-$\frac32$ modular forms ``associated'' with a given weight-$2$ modular form $f$ of level $N$, and relate its Fourier coefficients to central values of quadratic twists (real and imaginary) of $L(f,s)$. We will focus on examples for levels $N = 27$, $N = 15$, and $N=75$.

We enumerate all the spaces obtained by gluing in pairs the faces of the octahedron in an orientation-reversing fashion. Whenever such a gluing gives rise to nonmanifold points, we remove small open neighborhoods of these points, so we actually deal with three-dimensional manifolds with (possibly empty) boundary.

There are 298 combinatorially inequivalent gluing patterns, and we show that they define 191 distinct manifolds, of which 132 are hyperbolic and 59 are not. All the 132 hyperbolic manifolds have already been considered in different contexts by other authors, and we provide here their known “names” together with their main invariants. We also give the connected sum and JSJ decompositions for the 59 nonhyperbolic examples.

Our arguments make use of tools coming from hyperbolic geometry, together with quantum invariants and more classical techniques based on essential surfaces. Many (but not all) proofs were carried out by computer.

To study embeddings of tangles in knots, we use quandle cocycle invariants. Computations are carried out for tables of knots and tangles to investigate which tangles may or may not embed in knots in the tables.

Computing the relations for the coefficients satisfied by the characteristic polynomial of the Kronecker product of a general $n \times n$ matrix by a general $m \times m$ matrix leads to an elimination problem that is already difficult for small values of $n$ and $m$. In this article we focus on the problems for $(n, m) \in \{ (2,3), (2,4), (3,3)$ and use these problems for developing and testing a new elimination technique called elimination by degree steering.