Notes on Algebra and Geometry of Polynomial Representations
نویسنده
چکیده
Consider a semi-algebraic set A in R constructed from the sets which are determined by inequalities pi(x) > 0, pi(x) ≥ 0, or pi(x) = 0 for a given list of polynomials p1, . . . , pm. We prove several statements that fit into the following template. Assume that in a neighborhood of a boundary point the semi-algebraic set A can be described by an irreducible polynomial f . Then f is a factor of a certain multiplicity of some of the polynomials p1, . . . , pm. Special cases when A is elementary closed, elementary open, a polygon, or a polytope are considered separately. 2000 Mathematics Subject Classification. Primary: 14P10, Secondary: 52B05
منابع مشابه
CLUSTER ALGEBRAS AND CLUSTER CATEGORIES
These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...
متن کاملNotes on the Algebra and Geometry of Polynomial Representations
The paper deals with a semi-algebraic set A in R constructed by the inequalities pi(x) > 0, pi(x) ≥ 0, and pi(x) = 0 for a given list of polynomials p1, . . . , pm, and presents several statements that fit into the following template. Assume that in a neighborhood of a boundary point the semi-algebraic set A can be described by an irreducible polynomial f . Then f is a factor of a certain multi...
متن کاملNotes on representations of finite groups
1 Basic notions 4 1.1 ⊕, ⊗ and Hom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Invariant vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Invariant subspaces and irreducible representations . . . . . . . . . . . . . . . . . . . 5 1.4 The group algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...
متن کاملEEH: AGGH-like public key cryptosystem over the eisenstein integers using polynomial representations
GGH class of public-key cryptosystems relies on computational problems based on the closest vector problem (CVP) in lattices for their security. The subject of lattice based cryptography is very active and there have recently been new ideas that revolutionized the field. We present EEH, a GGH-Like public key cryptosystem based on the Eisenstein integers Z [ζ3] where ζ3 is a primitive...
متن کاملSome notes on the existence of an inequality in Banach algebra
We shall prove an existence inequality for two maps on Banach algebra, withan example and in sequel we have some results on R and Rn spaces. This waycan be applied for generalization of some subjects of mathematics in teachingwhich how we can extend a math problem to higher level.
متن کامل