Matroidal Root Structure of Skew Polynomials over Finite Fields
| Autor: | Felice Manganiello, Travis Baumbaugh |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Ring (mathematics)
Algebra and Number Theory Mathematics::Commutative Algebra Applied Mathematics Polynomial ring Skew Mathematics - Rings and Algebras 010103 numerical & computational mathematics 02 engineering and technology Divisibility rule 01 natural sciences Matroid Combinatorics Multiplication (music) Finite field Rings and Algebras (math.RA) 0202 electrical engineering electronic engineering information engineering FOS: Mathematics 020201 artificial intelligence & image processing Isomorphism 0101 mathematics Analysis Mathematics |
| DOI: | 10.48550/arxiv.1706.04283 |
| Popis: | A skew polynomial ring $R=K[x;\sigma,\delta]$ is a ring of polynomials with non-commutative multiplication. This creates a difference between left and right divisibility, and thus a concept of left and right evaluations and roots. A polynomial in such a ring may have more roots than its degree, which leads to the concepts of closures and independent sets of roots. There is also a structure of conjugacy classes on the roots. In $R=F_{q^m}[x,\sigma]$, this leads to matroids of right independent and left independent sets. These matroids are isomorphic via the extension of the map $\phi:[1]\to[1]$ defined by $\phi(a)=a^{\frac{q^{i-1}-1}{q-1}}$. Additionally, extending the field of coefficients of $R$ results in a new skew polynomial ring $S$ of which $R$ is a subring, and if the extension is taken to include roots of an evaluation polynomial of $f(x)$ (which does not depend on which side roots are being considered on), then all roots of $f(x)$ in $S$ are in the same conjugacy class. Comment: 23 pages |
| Databáze: | OpenAIRE |
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