polynomial basis

== English == === Noun === polynomial basis (plural polynomial bases) (algebra, ring theory) A basis of a polynomial ring (said ring being viewed either as a vector space over the field of coefficients or as a free module over the ring of coefficients). 2007, Nicos Karcanias, Efstathios Milonidis, 2: Structural Methods for Linear Systems: An Introduction, Matthew C. Turner, Declan G. Bates (editors), Mathematical Methods for Robust and Nonlinear Control: EPSRC Summer School, Springer, Lecture Notes in Control and Information Sciences 367, page 89, If T ( s ) = M ( s ) D ( s ) − 1 {\displaystyle T(s)=M(s)D(s)^{-1}} is a RCMFD[right coprime matrix factor description] of T ( s ) {\displaystyle T(s)} , then M ( s ) {\displaystyle M(s)} is a polynomial basis for X t {\displaystyle {\mathfrak {X}}_{t}} . If Q ( s ) {\displaystyle Q(s)} is a greatest right divisor of M ( s ) {\displaystyle M(s)} then T ( s ) = M ¯ ( s ) Q ( s ) D ( s ) − 1 {\displaystyle T(s)={\overline {M}}(s)Q(s)D(s)^{-1}} , where M ¯ ( s ) {\displaystyle {\overline {M}}(s)} is a least degree polynomial basis of X t {\displaystyle {\mathfrak {X}}_{t}} [15]. (algebra, field theory, cryptography, of a finite field) Specifically, a basis, of the form {1, α, ..., αn-1}, of a finite extension Fqn of a Galois field Fq, where α is a primitive element of Fqn (i.e., a root of a degree-n primitive polynomial over Fq). 2010, Vladimir Tujillo-Olaya, Jaime Velasco-Medina, Hardware Architectures for Elliptic Curve Cryptoprocessors Using Polynomial and Gaussian Normal Basis Over GF(2233), Marina L. Gravrilova, C. J. Kenneth Tan, Edward David Moreno (editors), Transactions on Computational Science XI: Special Issue on Security in Computing, Part 2, Springer, LNCS 6480, page 79, In this case, the GF(2m) multiplication is implemented in hardware using three algorithms for polynomial basis (PB) and three for gaussian normal basis (GNB). ==== Hyponyms ==== (basis of a polynomial ring): Bernstein basis, monomial basis ==== Translations ==== === See also === Bernstein basis polynomial dual basis monomial normal basis === Further reading === Basis (linear algebra) on Wikipedia.Wikipedia Dual basis on Wikipedia.Wikipedia Normal basis on Wikipedia.Wikipedia Orthogonal polynomials on Wikipedia.Wikipedia Change of basis on Wikipedia.Wikipedia Basis function on Wikipedia.Wikipedia