Laurent polynomial - Wikipedia
WEBIn mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field is a linear combination of positive and negative powers of the variable with coefficients in . Laurent polynomials in X {\displaystyle X} form a ring denoted F [ X , X − 1 ] {\displaystyle \mathbb {F} [X,X^{-1}]} . [ 1 ]
Laurent Polynomial -- from Wolfram MathWorld
WEBAug 22, 2024 · A Laurent polynomial with coefficients in the field F is an algebraic object that is typically expressed in the form ...+a_(-n)t^(-n)+a_(-(n-1))t^(-(n-1))+... +a_(-1)t^(-1)+a_0+a_1t+...+a_nt^n+..., where the a_i are elements of F, and only finitely many of the a_i are nonzero.
Laurent series - Wikipedia
WEBA Laurent polynomial is a Laurent series in which only finitely many coefficients are non-zero. Laurent polynomials differ from ordinary polynomials in that they may have terms of negative degree.
How to prove the ring of Laurent polynomials over a field is a ...
WEBGiven a Laurent polynomial f ∈ F[x, x − 1], define its "negative degree" deg − (f) to be the largest power of x − 1 that appears in f. Let I be an ideal of F[x, x − 1]. Note that {x − deg − (f) f ∣ f ∈ I} ⊆ F[x]. Let J be the ideal in F[x] generated by this set.
Laurent Polynomials Ring - Mathematics Stack Exchange
WEBWikipedia says: "The Laurent polynomial ring $R[X, X^{−1}]$ is isomorphic to the group ring of the group $\mathbb{Z}$ of integers over $R$". Can anyone offer a proof? I also don't fully understand what a group ring is, even after reading the wiki article.
What are the units in the ring of Laurent polynomials?
WEBWhat are the units in $R[X,X^{-1}]$, where $R$ is a commutative ring with $1$? I know that the question for polynomial rings is a standard textbook exercise. However, I couldn't find a reference for Laurent polynomials, since most people only seem to consider coefficients in an integral domain.
WEBIn this paper we study a situation which lies between the polynomial and rational function cases: namely, we study Laurent polynomials, i.e., ratio- nal functions of the form f(x)=xnwith f 2C[x]. We will prove that de- compositions of Laurent polynomials satisfy variants of Ritt’s results.
WEBW e abbreviate L 0 = ( t ). Note that is a unique factorization domain, so an y t w o elemen ts x; y 2 L 0 ha v e a ell-de ned greatest common divisor gcd( ) whic h is an elemen t of L 0 de ned up to a m ultiple from the group. 0 in v ertible elemen ts in L 0 ; the group.
WEBThe Jones polynomial of a knot (and generally a link with an odd number of components) is a Laurent polynomial in t. The most elementary ways to calculate V L (t) use the “linear skein theory”
Laurent polynomial in nLab
WEBGiven indeterminates t 1, …, t k t_1,\ldots, t_k a Laurent polynomial over a field or ring F F is a polynomial in t 1, …, t k, t 1 − 1, …, t k − 1 t_1,\ldots,t_k,t_1^{-1},\ldots,t_k^{-1} over F F.