polynomial basis

polynomial basis (plural polynomial bases)

1. (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 ${\displaystyle T(s)=M(s)D(s)^{-1}}$  is a RCMFD^{[right coprime matrix factor description]} of ${\displaystyle T(s)}$ , then ${\displaystyle M(s)}$  is a polynomial basis for ${\displaystyle {\mathfrak {X}}_{t}}$ . If ${\displaystyle Q(s)}$  is a greatest right divisor of ${\displaystyle M(s)}$  then ${\displaystyle T(s)={\overline {M}}(s)Q(s)D(s)^{-1}}$ , where ${\displaystyle {\overline {M}}(s)}$  is a least degree polynomial basis of ${\displaystyle {\mathfrak {X}}_{t}}$  [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 ${\displaystyle f}$  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”, in 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.

2. (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 ${\displaystyle \mathbb {F} _{2^{n}}}$  is the multiplication of two polynomials of degree at most ${\displaystyle n-1}$  in ${\displaystyle \mathbb {F} _{2}[x]}$ .

   • 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(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^[1], 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 ${\displaystyle \theta }$  be a root of a monic irreducible polynomial ${\displaystyle f}$  of degree ${\displaystyle m}$  over ${\displaystyle K=\mathbb {F} _{q}}$ , and let ${\displaystyle B=\{1,\theta ,\theta ^{2},\dots \theta ^{m-1}\}}$  be the corresponding polynomial basis of ${\displaystyle F=\mathbb {F} _{q^{m}}}$  over ${\displaystyle K}$ . Then the dual basis ${\displaystyle B^{*}}$  of ${\displaystyle B}$  is a polynomial basis if and only if ${\displaystyle f}$  is a binomial and ${\displaystyle m\equiv 1\!\!\!\!{\pmod {p}}}$ , where ${\displaystyle q}$  is a power of the prime ${\displaystyle p}$ .
     5.1.14 Corollary There exists a dual pair of polynomial bases of ${\displaystyle \mathbb {F} _{q^{m}}}$  over ${\displaystyle \mathbb {F} _{q}}$  if and only if the following three conditions are satisfied: […] .

• (basis of a polynomial ring): Bernstein basis, monomial basis

• Bernstein basis polynomial
• dual basis
• monomial
• normal basis

•   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