Related papers: On Canonical Subfield Preserving Polynomials
For the space of single-variable monic and centered complex polynomial vector fields of arbitrary degree d, it is proved that any bifurcation which preserves the multiplicity of equilibrium points can be realized as a composition of a…
We use generating functions over group rings to count polynomials over finite fields with the first few coefficients prescribed and a factorization pattern prescribed. In particular, we obtain different exact formulas for the number of…
Let $K$ be an algebraically closed field with an absolute value. This note gives an elementary proof of the classical result that the roots of a polynomial with coefficients in $K$ are continuous functions of the coefficients of the…
Recently, there has been renewed interest in the theory and applications of de Paiva's dialectica categories and their relationship to the category of polynomial functors. Both fall under the theory of generalized polynomial categories,…
We study the existence and left properness of transferred model structures for "monoid-like" objects in monoidal model categories. These include genuine monoids, but also all kinds of operads as for instance symmetric, cyclic, modular,…
The class of self-conjugate-reciprocal irreducible monic (SCRIM) polynomials over finite fields are studied. Necessary and sufficient conditions for monic irreducible polynomials to be SCRIM are given. The number of SCRIM polynomials of a…
We construct an effective algorithmic method to compute the homological monodromy of a complex polynomial which is tame. As an application we show the existence of conjugated polynomials in a number field which are not topologically…
In this document, we collect a list of categorical structures on the category $\mathbf{Poly}$ of polynomial functors. There is no implied claim that this list is in any way complete. It includes: infinitely many monoidal structures, all but…
Using the formalism of polynomials with positive coefficients, the fact that exactly half of all subsets of a finite set have even cardinality can be generalized asymptotically.
In this paper I consider all possible properties from commutative algebra for polynomial composites and monoid domains. The aim is full characterization of these structures. I start with the examination of group, ring, modules properties,…
Let $\A$ be an algebra and let $f(x_1,...,x_d)$ be a multilinear polynomial in noncommuting indeterminates $x_i$. We consider the problem of describing linear maps $\phi:\A\to \A$ that preserve zeros of $f$. Under certain technical…
Canonical matrices are given for (a) bilinear forms over an algebraically closed or real closed field; (b) sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution; and (c) sesquilinear…
We study the action of monads on categories equipped with several monoidal structures. We identify the structure and conditions that guarantee that the higher monoidal structure is inherited by the category of algebras over the monad.…
Most results on the value sets $V_f$ of polynomials $f \in \mathbb{F}_q[x]$ relate the cardinality $|V_f|$ to the degree of $f$. In particular, the structure of the spectrum of the class of polynomials of a fixed degree $d$ is rather well…
It is well-known that the category of Kleisli algebras for a monoidal monad carries a canonical monoidal structure. We define the notion of a commutative graded monad and present a strictly two-categorical proof that Kleisli algebras for…
In this paper, as an extension of the integer case, we define polynomial functions over the residue class rings of Dedekind domains, and then we give canonical representations and counting formulas for such polynomial functions. In…
A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we enumerate self-reciprocal irreducible monic polynomials over a finite field with prescribed leading coefficients.…
We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by…
In this technical report we describe a general class of monoids for which (sub)sequential rational can be characterised in terms of a congruence relation in the flavour of Myhill-Nerode relation. The class of monoids that we consider can be…
We construct free monoids in a monoidal category with finite limits and countable colimits, in which tensoring on either side preserves reflexive coequalizers and colimits of countable chains.