polynomial basis
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English[edit]
Noun[edit]
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 is a RCMFD^{[right coprime matrix factor description]} of , then is a polynomial basis for . If is a greatest right divisor of then , where is a least degree polynomial basis of [15].
- 2009, Jan Flusser, Barbara Zitova, Tomas Suk, Moments and Moment Invariants in Pattern Recognition, Wiley, page 166,
- When dealing with various polynomial bases up to a certain degree and with corresponding moments, any moment (with respect to any basis) can be expressed as a function of moments of the same or fewer orders with respect to an arbitrary basis. […] As we already saw in Chapter 1, OG^{[orthogonal]} moments are, unlike geometric and all other moments, coordinates of in the polynomial basis in the common sense used in linear algebra.
- 2012, A. Kominek, et al., Ident. of Low-Compl. LPV-IO Models for Engine Control, Javad Mohammadpour, Carsten W. Scherer (editors), Control of Linear Parameter Varying Systems with Applications, Springer, page 452,
- For this purpose, a polynomial basis is used here, which consists of all monomials in the scheduling signals up to a fixed total order. Such a polynomial basis can be interpreted as the polynomial terms, which would appear in a multivariate Taylor approximation of the unknown scheduling function.
- 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,
- (algebra, field theory, cryptography, of a finite field) Specifically, a basis, of the form {1, α, ..., α^{n-1}}, of a finite extension F_{qn} of a Galois field F_{q}, where α is a primitive element of F_{qn} (i.e., a root of a degree-n primitive polynomial over F_{q}).
- 1999, I. Blake, G. Seroussi, N. Smart, Elliptic Curves in Cryptography, Cambridge University Press, page 20,
- When using polynomial bases, the first stage in computing the product of two elements of is the multiplication of two polynomials of degree at most in .
- 2010, Vladimir Tujillo-Olaya, Jaime Velasco-Medina, Hardware Architectures for Elliptic Curve Cryptoprocessors Using Polynomial and Gaussian Normal Basis Over GF(2^{233}), 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(2^{m}) multiplication is implemented in hardware using three algorithms for polynomial basis (PB) and three for gaussian normal basis (GNB).
- 2013, Gary L. Mullen, Daniel Panario, Handbook of Finite Fields, Taylor & Francis (Chapman & Hall / CRC Press), page 103,
- In contrast to the case of normal bases considered in Theorem 5.1.9, the dual basis of a polynomial basis is usually not a polynomial basis.
- […]
- 5.1.13 Theorem [1265, 1298] Let be a root of a monic irreducible polynomial of degree over , and let be the corresponding polynomial basis of over . Then the dual basis of is a polynomial basis if and only if is a binomial and , where is a power of the prime .
- 5.1.14 Corollary There exists a dual pair of polynomial bases of over if and only if the following three conditions are satisfied: […] .
- 1999, I. Blake, G. Seroussi, N. Smart, Elliptic Curves in Cryptography, Cambridge University Press, page 20,
Hyponyms[edit]
- (basis of a polynomial ring): Bernstein basis, monomial basis
Translations[edit]
basis of a polynomial ring
See also[edit]
Further reading[edit]
- 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