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We present a construction of new invariant sets for fibred polynomial dynamics with base an irrational rotation over the unit circle, called multi-curves. Furthermore, the local dynamical theory for attracting invariant curves is extended…
We study the concept of quadratic variation of a continuous path along a sequence of partitions and its dependence with respect to the choice of the partition sequence. We define the concept of quadratic roughness of a path along a…
An infinite family of quasi-maximally superintegrable Hamiltonians with a common set of (2N-3) integrals of the motion is introduced. The integrability properties of all these Hamiltonians are shown to be a consequence of a hidden…
Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables.…
We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds…
Planar polynomial automorphisms are polynomial maps of the plane whose inverse is also a polynomial map. A map is reversible if it is conjugate to its inverse. Here we obtain a normal form for automorphisms that are reversible by an…
In this paper we introduce and discuss some classes of orthogonal polynomials in several non-commuting variables. The emphasis is on a non-commutative version of the orthogonal polynomials on the real line. We introduce recurrence equations…
A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^{\alpha}\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…
It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…
The mathematical notion of foliated cobordism is presented, and its relationship to both the motion of extended particles and wave motion is detailed. The fact that wave motion, when represented in such a manner on a four-dimensional…
Dynamical systems with quadratic or polynomial drift exhibit complex dynamics, yet compared to nonlinear systems in general form, are often easier to analyze, simulate, control, and learn. Results going back over a century have shown that…
The linearization of a quadratic form gives rise to a Clifford algebra structure, as seen in Dirac's factorization of the d'Alembert operator. A similar structure known as a generalized Clifford algebra arises from the continuation of this…
We introduce G{\aa}rding polynomials, a class of real multivariate polynomials characterized by positivity regions that are invariant under translation by positive vectors and closed under strictly positive affine transformations. We prove…
The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviors and, conversely, starting from a desired behavior allows the design of objects exhibiting that behavior in engineering.…
Orthogonal polynomials in two variables on cubic curves are considered, including the case of elliptic curves. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal…
Some cubic polynomials over the integers have three distinct real roots with continued fractions that all have the same common tail. We characterize the polynomials for which this happens, and then investigate the situation for other…
We give an efficient algorithm to enumerate all sets of $r\ge 1$ quadratic polynomials over a finite field, which remain irreducible under iterations and compositions.
We prove that non-hyperbolic non-renormalizable quadratic polynomials are expansion inducing. For renormalizable polynomials a counterpart of this statement is that in the case of unbounded combinatorics renormalized mappings become almost…
The restricted planar four body problem describes the motion of a massless body under the Newtonian gravitational force of other three bodies (the primaries), of which the motion gives us general solutions of the three body problem. A…