Polynomial ring

Summary

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings. A closely related notion is that of the ring of pol

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https://en.wikipedia.org/wiki/Polynomial_ring

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Vinberg's theta-groups in positive characteristic and Kostant-Weierstrass slices

We generalize the basic results of Vinberg's theta-groups, or periodically graded reductive Lie algebras, to fields of good positive characteristic. To this end we clarify the relationship between the little Weyl group and the (standard) Weyl group. We deduce that the ring of invariants associated to the grading is a polynomial ring. This approach allows us to prove the existence of a KW-section for a classical graded Lie algebra (in zero or odd positive characteristic), confirming a conjecture of Popov in this case.

Birkhäuser-Verlag, Springer2009

Arithmetic Bounds-Lenstra's Constant and Torsion of K-Groups

Julien Houriet

This thesis is concerned with computations of bounds for two different arithmetic invariants. In both cases it is done with the intention of proving some algebraic or arithmetic properties for number fields. The first part is devoted to computations of lower bounds for the Lenstra's constant. For a number field K the Lenstra's constant is denoted Λ(K) and defined as the length of the largest exceptional sequence in K. An exceptional sequence is a set of units in K such that for any two among them their difference is a unit as well. H.W. Lenstra showed that if Λ(K) is large enough – bigger than a constant depending on the degree and the discriminant of K – then the ring of integers of K is Euclidean with respect to the norm. Using computer software PARI/GP and some algorithms from graph theory we construct exceptional sequences in number fields having a small discriminant. These exceptional sequences yield lower bounds for Lenstra's constant which are large enough to prove the existence of 42 new Euclidean number fields of degree 8 to 12. The aim of the second part of this thesis is proving upper bounds for the torsion part of the K-groups of a number field ring of integers. A method due to C. Soulé yields bounds for the torsion of these K-groups depending on an invariant of hermitian lattices over number fields. Firstly we describe some properties of rank one hermitian lattices, especially of ideal lattices. Secondly we apply these properties to arbitrary rank hermitian lattices and this implies a significant improvement of the upper bounds for their invariants and accordingly for the torsion of K-groups. The progress mainly achieves much lower contributions of the number field attributes, particularly the degree and the absolute discriminant.

EPFL2010

Variants of Homomorphism Polynomials Complete for Algebraic Complexity Classes

Aditya Vardhan Varre

We present polynomial families complete for the well-studied algebraic complexity classes VF, VBP, VP, and VNP. The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. (2018). We consider three different variants of graph homomorphisms, namely injective homomorphisms, directed homomorphisms, and injective directed homomorphisms, and obtain polynomial families complete for VF, VBP, VP, and VNP under each one of these. The polynomial families have the following properties: The polynomial families complete for VF, VBP, and VP are model independent, i.e., they do not use a particular instance of a formula, algebraic branching programs, or circuit for characterising VF, VBP, or VP, respectively. All the polynomial families are hard under p-projections.

ASSOC COMPUTING MACHINERY2021