# polynomial basis

## English

### Noun

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, 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, ..., 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).
• 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(2m) 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.
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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: [] .