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Laurent polynomial

A Laurent polynomial is a generalization of an ordinary polynomial that allows for negative integer exponents in the powers of the indeterminate, forming a finite sum \sum_{k=m}^{n} a_k t^k where m, n \in \mathbb{Z}, m \leq n, the coefficients a_k belong to a given field F, and only finitely many a_k are nonzero. This structure extends the polynomial ring F by including inverses of t, resulting in the Laurent polynomial ring F[t, t^{-1}], which is the localization of F at the multiplicative set generated by t. The ring operations are defined componentwise for addition and via the rule t^k \cdot t^l = t^{k+l} for multiplication, making it an integral domain if F is a field. Laurent polynomials can be defined in multiple variables as elements of F[t_1^{\pm 1}, \dots, t_k^{\pm 1}], where each indeterminate and its inverse are included, and they form a under the standard operations. This multivariate version arises naturally as the F[\mathbb{Z}^k], associating to the on k generators. They are distinct from , which allow infinitely many negative powers, but Laurent polynomials serve as the algebraic foundation for studying such series in and . Named after the French mathematician Pierre Alphonse Laurent (1813–1854), who introduced the related concept of in his 1843 memoir on the , these objects play key roles in diverse areas including toric geometry, where they parameterize torus actions, and , where mutations preserve Laurent polynomial structure. In signal processing, Laurent polynomials model filters.

Fundamentals

Definition

A Laurent polynomial in one indeterminate x over a R with is an expression of the form \sum_{k=m}^{n} a_k x^k, where m, n \in \mathbb{Z} with m \leq n, the coefficients a_k \in R, and only finitely many of the a_k are nonzero. The of such a Laurent polynomial, defined as the set of exponents k for which a_k \neq 0, is therefore a finite of \mathbb{Z} that is bounded both above and below. This construction generalizes to multiple indeterminates x_1, \dots, x_d, where a Laurent polynomial takes the form of a finite sum \sum_{\alpha} a_{\alpha} x^{\alpha} with \alpha \in \mathbb{Z}^d and finite support, belonging to the localization R[x_1, \dots, x_d, x_1^{-1}, \dots, x_d^{-1}]. The ring of Laurent polynomials in one indeterminate, denoted R[x, x^{-1}], is the localization of the polynomial ring R at the multiplicative set consisting of the powers of x.

Notation and Examples

Laurent polynomials in one variable over a R (such as \mathbb{Z}, \mathbb{Q}, or \mathbb{R}) are commonly denoted as formal sums p(x) = \sum_{k \in \mathbb{Z}} a_k x^k, where a_k \in R and only finitely many a_k are nonzero. The exponents k range over all integers, allowing negative powers, and the summation is typically ordered from the lowest to the highest exponent for clarity, such as p(x) = \sum_{k=m}^n a_k x^k with m \leq n and m possibly negative. A simple example is the Laurent polynomial x^{-1} + 1 + x over \mathbb{[Q](/page/Q)}, which includes a negative exponent term and distinguishes it from ordinary . For a more complex case over \mathbb{R}, consider $2x^{-2} + 3x + 5x^3, featuring multiple negative and positive powers with real coefficients. In multiple variables, Laurent polynomials extend similarly; for instance, over \mathbb{[Q](/page/Q)}, f(x,y) = x^{-1} y + 1 + x y^2 incorporates negative exponents in one variable while allowing positive or zero in the other. To illustrate their structure, consider the multiplication (x^{-1} + x)(1 + x^2): \begin{align*} (x^{-1} + x)(1 + x^2) &= x^{-1} \cdot 1 + x^{-1} \cdot x^2 + x \cdot 1 + x \cdot x^2 \\ &= x^{-1} + x + x + x^3 \\ &= x^{-1} + 2x + x^3. \end{align*} This computation demonstrates how negative exponents combine via the rule x^a \cdot x^b = x^{a+b} for a, b \in \mathbb{Z}.

Algebraic Structure

Ring Operations

Laurent polynomials over a R form a under the operations of and , denoted R[x, x^{-1}]. is performed component-wise on the coefficients, preserving the finite property. For two Laurent polynomials f = \sum_{k \in \mathbb{Z}} a_k x^k and g = \sum_{k \in \mathbb{Z}} b_k x^k, where only finitely many a_k and b_k are nonzero, their is f + g = \sum_{k \in \mathbb{Z}} (a_k + b_k) x^k. This operation is associative and commutative because in R is, and the zero element serves as the , defined as the Laurent with all coefficients zero. Multiplication extends the to accommodate negative exponents, again ensuring finite support in the result due to the finite number of nonzero terms in each factor. Specifically, f \cdot g = \sum_{n \in \mathbb{Z}} c_n x^n, where c_n = \sum_{k + m = n} a_k b_m, and the inner sum is finite. This multiplication is associative and distributive over addition, inheriting these properties from the ring structure of R, and it is commutative if R is. The multiplicative identity is the $1 = x^0, with 1 at exponent 0 and zeros elsewhere. The set R[x, x^{-1}] is closed under these operations: combines finite s into another finite , while yields a finite because the possible exponents n over a finite determined by the minimal and maximal exponents of f and g. Together with the identities and the inherited algebraic properties from R, this endows R[x, x^{-1}] with the structure of a with unity. There is a natural ring homomorphism from the polynomial ring R to R[x, x^{-1}], given by the inclusion map that embeds polynomials (with nonnegative exponents) as a subring, preserving addition and multiplication.

Module Properties

Laurent polynomials over a commutative ring R form the ring R[x, x^{-1}], which is a free R-module with basis \{x^k \mid k \in \mathbb{Z}\}. Each element is a finite sum \sum_{k \in \mathbb{Z}} a_k x^k with a_k \in R and only finitely many a_k nonzero, ensuring the uniqueness of this representation due to the linear independence of the basis elements. This free module structure implies that R[x, x^{-1}] is projective and flat over R, facilitating computations in homological algebra. From a localization viewpoint, R[x, x^{-1}] is isomorphic to the localization of the R at the multiplicative set S = \{1, x, x^2, \dots \}, denoted S^{-1}R. This construction allows Laurent polynomial modules to inherit key properties from modules, such as coherence when R is Noetherian, and provides a framework for inverting powers of x while preserving the finite support condition. The localization map embeds R into R[x, x^{-1}], with elements of the form f(x)/x^n for f \in R and n \geq 0 corresponding exactly to Laurent polynomials. As a \mathbb{Z}-graded module over itself, R[x, x^{-1}] admits the grading \deg(x^k) = k, yielding the direct sum decomposition R[x, x^{-1}] = \bigoplus_{k \in \mathbb{Z}} R \cdot x^k, where each homogeneous component R \cdot x^k \cong R is free of rank one over R. This grading extends to modules over R[x, x^{-1}], enabling the study of graded resolutions and filtrations. For instance, the associated graded ring in certain filtrations aligns with representations of algebras, preserving the direct sum structure. Modules over Laurent polynomial rings participate in tensor products and exact sequences with computable derived functors in simple cases. Since finitely generated projective modules over R[x, x^{-1}] (with R a PID or field) are free, higher Tor groups vanish for such modules, reflecting exactness in tensor products with free resolutions. Similarly, Ext groups measure extensions; for example, in A-hypergeometric systems defined by Laurent polynomials, explicit computations show that \operatorname{Ext}^i and \operatorname{Tor}_i groups against parameter sheaves are supported in low degrees, often zero beyond i=1 for toric ideals. These properties arise from the free basis and grading, ensuring short exact sequences split in projective cases.

Key Properties

Degree and Valuation

For a nonzero Laurent polynomial p = \sum_{k \in \mathbb{Z}} a_k x^k with finitely many nonzero coefficients a_k in a ring R, the degree, denoted \deg(p), is defined as the maximum exponent k such that a_k \neq 0; the zero polynomial is assigned degree -\infty. Similarly, the valuation, denoted v(p), is the minimum exponent k such that a_k \neq 0; the zero polynomial has valuation +\infty. These measures quantify the span of the support of p, distinguishing Laurent polynomials from ordinary polynomials where valuations are nonnegative. Under addition, the satisfies \deg(p + q) \leq \max(\deg p, \deg q), with holding if \deg p \neq \deg q or if the leading coefficients do not cancel. Analogously, the valuation obeys v(p + q) \geq \min(v p, v q), with strict under the corresponding non-cancellation condition for the lowest-degree terms. For multiplication, both functions are additive: \deg(pq) = \deg p + \deg q and v(pq) = v p + v q, as the leading and lowest terms multiply without overlap or cancellation. The leading coefficient of a nonzero Laurent polynomial p is the coefficient a_{\deg(p)} of its highest-degree term. If R is a , the content of p is the of all its coefficients \{a_k\}. These allow normalization, such as making the leading coefficient a unit or extracting the content to obtain a primitive polynomial. For example, consider p = x^{-2} + [3x](/page/3X) + 1. Here, the lowest-degree term is x^{-2} with 1, so v(p) = -2; the highest-degree term is [3x](/page/3X) with 3, so \deg(p) = 1 and the leading is 3; the is \gcd(1, 3, 1) = 1.

Units and Divisibility

In the ring of Laurent polynomials [R](/page/R)[x, x^{-1}], where [R](/page/R) is a with identity, a is an element u for which there exists v such that u v = 1. These s are precisely the monomials a x^k, where a is a in [R](/page/R) and k \in \mathbb{Z}. This characterization follows from the additivity properties of the valuation v(u), defined as the minimal exponent with nonzero , and the s(u) = \deg(u) - v(u), where \deg(u) is the maximal exponent with nonzero . For u v = 1, we have v(u) + v(v) = 0 and s(u) + s(v) = 0, implying s(u) = 0, so u (and v) must be monomials. The a of u must then satisfy a b = 1 for some b \in R, making a a unit in R. Divisibility in R[x, x^{-1}] is defined in the standard way: p divides q if there exists r such that q = p r. When R = F is a , the univariate Laurent polynomial ring F[x, x^{-1}] is a , meaning every nonzero ideal is generated by a single element. This structure admits an adapted based on the valuation v. For nonzero f, g \in F[x, x^{-1}] with v(g) \leq v(f), there exist q, r such that f = q g + r with either r = 0 or v(r) > v(g); specifically, shift by multiplying by x^{-v(g)} to reduce to the polynomial case in F, perform division there, and shift back. Iterating this yields the via the usual process.

Relations to Other Mathematical Objects

Comparison with Polynomials

Laurent polynomials and ordinary polynomials share fundamental algebraic structures, particularly when considering elements without negative exponents. Both form when the coefficient R is an , supporting the same and operations for terms with nonnegative powers of the indeterminate x. In this sense, the of polynomials R behaves identically to the of Laurent polynomials consisting solely of nonnegative powers. A key difference lies in the allowed exponents: ordinary polynomials in R are finite sums \sum_{i=0}^n a_i x^i with a_i \in R and nonnegative integers i, whereas Laurent polynomials in R[x, x^{-1}] permit integer exponents, including negative ones, as in \sum_{i=m}^n a_i x^i where m \in \mathbb{Z} may be negative. This extension enables the inclusion of inverses like x^{-1}, which is not an element of R but is a Laurent monomial. Consequently, R embeds naturally as a subring into R[x, x^{-1}], specifically as the localization of R at the multiplicative set S = \{1, x, x^2, \dots \}. The R is not closed under inversion for nonconstant elements, limiting it to constant units; for instance, x has no in R. In contrast, Laurent polynomials bridge this gap by making monomials c x^k (with c \in R^\times) units, since (c x^k) \cdot (c^{-1} x^{-k}) = 1. This richer unit group facilitates connections to the quotient field R(x) of rational functions, which coincides for both rings, as Laurent polynomials already incorporate x^{-1}.

Connection to Laurent Series

Laurent series are formal sums of the form \sum_{k=-\infty}^{\infty} a_k x^k, where the coefficients a_k belong to a commutative ring R, but with the restriction that only finitely many negative powers have nonzero coefficients, allowing the series to extend infinitely in the positive direction without finiteness requirements on the support. In contrast to this potentially infinite extension, Laurent polynomials form a proper subset of the ring of formal Laurent series, consisting precisely of those series with finite support—meaning only finitely many coefficients a_k are nonzero overall, both for positive and negative exponents. The ring of Laurent polynomials thus embeds naturally into the larger ring of formal Laurent series over R, denoted R((x)), which generalizes formal power series R[] by permitting a finite principal part of negative powers. This inclusion respects the algebraic structure: the polynomials in R sit inside the Laurent polynomials R[x, x^{-1}], which in turn sit inside the formal Laurent series R((x)), all as subrings of formal power series in x and x^{-1}. Addition and multiplication operations on Laurent polynomials coincide exactly with those defined on the ambient ring of formal Laurent series, wherever they are applicable, preserving the commutative ring structure without issue. While Laurent polynomials are treated purely formally as algebraic objects, the broader context of Laurent series often extends to analytic settings, where convergence is considered in an annular region around a point in the complex plane, though this analytic property does not apply to the finite-support case of polynomials.

Applications

In Knot Theory and Combinatorics

Laurent polynomials play a central role in as invariants that distinguish knots and links up to . The , introduced by J. W. Alexander in 1923, is a fundamental example of such an , assigning to each oriented or link a Laurent in one variable t with integer coefficients, unique up to multiplication by units \pm t^k in the Laurent \mathbb{Z}[t, t^{-1}]. It can be defined combinatorially using the Seifert matrix associated to a Seifert surface of the , where the is the of t^{1/2} V - t^{-1/2} V^T for the Seifert matrix V, or algebraically via Fox calculus applied to a of the , yielding the Alexander matrix whose (1,1)- generates the ideal of the Alexander module. A representative example is the , whose is t^{-1} - 1 + t. The exhibits key properties, including multiplicativity under connected sum of knots, so that \Delta_{K \# L}(t) = \Delta_K(t) \Delta_L(t), and invariance under , which implies it remains unchanged under Reidemeister moves on knot diagrams. Another prominent Laurent polynomial invariant is the Jones polynomial, discovered by V. F. R. Jones in 1984, which assigns to each oriented link a Laurent polynomial in the variable A over \mathbb{Z}[A, A^{-1}]. It satisfies the skein relation A L_+ - A^{-1} L_- = (A^2 - A^{-2}) L_0 for crossings differing locally, with normalization on the unknot, and can be derived combinatorially from the Kauffman bracket, a state sum over link diagrams where each state contributes a factor of A^{\alpha(s) - \beta(s)} (-A^2 - A^{-2})^{o(s) - 1}, with \alpha(s), \beta(s), and o(s) counting A-smoothings, B-smoothings, and circles in the state s, respectively; the Jones polynomial is then obtained by multiplying the bracket by (-A^3)^{-w(D)} for writhe w(D). Like the , the Jones polynomial is multiplicative under connected sum and invariant under Reidemeister moves. In enumerative combinatorics, Laurent polynomials arise as generating functions for combinatorial objects related to knot invariants, such as paths and tilings encoded in algebraic structures like the Temperley-Lieb algebra, where basis elements correspond to non-crossing matchings or planar diagrams, and traces yield evaluations of polynomials like the Jones invariant that count weighted paths or tiling configurations.

In Algebraic Geometry and Signal Processing

In algebraic geometry, Laurent polynomials play a central role in defining hypersurfaces within the algebraic torus (\mathbb{C}^*)^n, where the coordinate ring is the ring of Laurent polynomials \mathbb{C}[z_1^{\pm 1}, \dots, z_n^{\pm 1}]. These hypersurfaces arise as zero loci V(f) of nonzero Laurent polynomials f, and their closures in toric varieties provide models for studying geometric properties like intersections and volumes. In toric varieties, which generalize projective and affine spaces, affine patches are spectra of semigroup rings over dual cones, and hypersurfaces defined by Laurent equations correspond to divisors supported on torus-invariant subvarieties. The polytope of a Laurent polynomial f, defined as the of the exponents in its , associates geometric structure to these hypersurfaces and links them to —the images under the map \Log: (\mathbb{C}^*)^n \to \mathbb{R}^n, z \mapsto (\log |z_1|, \dots, \log |z_n|). Amoebas of hypersurfaces in toric varieties connect to , where the complement of the amoeba consists of connected components corresponding to Laurent monomials not in the tropicalization, and the number of components is bounded by the mixed volume of the polytopes. This association facilitates the study of singularities and enumerative invariants in toric settings. Modules over Laurent polynomial rings model quasicoherent sheaves on the and its compactifications in toric varieties, where graded modules over the total coordinate ring correspond to such sheaves via the structure sheaf. In the cohomology of torus actions on toric varieties, these modules inform equivariant computations, as the cohomology ring is isomorphic to the Stanley-Reisner ring of the , with Laurent monomials generating sections of line bundles. Finitely generated torsion-free modules over \mathbb{Z}[t, t^{-1}] are classified by sub-lattices, with their orders given by determinants of endomorphisms, aiding in understanding torus representations. Resolution of singularities in toric varieties employs to identify singular , resolved via toric blowups corresponding to star subdivisions of the . For instance, the of the y^2 = x^5 in \mathbb{C}^2 is resolved by subdividing the maximal , yielding a toric surface where the exceptional is a torus-invariant . This process refines the to ensure all cones are , preserving the toric structure while desingularizing the . In , Laurent polynomials model filters in multirate systems, particularly for perfect s in theory. A two-channel uses analysis filters H_0(z), H_1(z) and filters G_0(z), G_1(z), all Laurent polynomials, satisfying the perfect condition H_0(z)G_0(z) + H_1(z)G_1(z) = 2z^{-l} for some delay l, and alias cancellation H_0(z)G_0(-z) + H_1(z)G_1(-z) = 0. This enables decomposition and of signals without loss, as in Daubechies wavelets where lowpass filters are designed via spectral factorization of Laurent polynomials. The Quillen-Suslin theorem ensures the existence of FIR filters for multidimensional non-redundant banks by representing unimodular matrices over Laurent rings. MATLAB's filters2lp function converts filters to Laurent polynomials, facilitating verification of ; for the biorthogonal bior1.3, it yields lowpass LoDz(z) and LoRz(z) polynomials such that LoRz(z)LoDz(z) + HiRz(z)HiDz(z) = 2, confirming exact . Orthogonal Laurent polynomials (OLPs), obtained via Gram-Schmidt orthogonalization of the basis \{z^{-k}, z^k \mid k \geq 0\} with respect to a positive measure on the , address problems on unbounded intervals. For a bisequence of moments \{\mu_n\}_{n=-\infty}^\infty, the strong seeks a whose moments match \mu_n = \int z^n d\psi(z), solved using continued fractions from OLP recurrences and Nevanlinna parametrization for uniqueness. OLPs also enable formulas approximating integrals \int_a^b f(x) w(x) dx \approx \sum_{k=1}^n \lambda_k^{(n)} f(t_k^{(n)}), where nodes t_k^{(n)} are OLP zeros and weights \lambda_k^{(n)} derive from the Christoffel-Darboux kernel, converging for regular measures. Regular OLPs satisfy a three-term recurrence Q_n(z) = (z - \beta_n) Q_{n-1}(z) - \alpha_n z Q_{n-2}(z) with positive coefficients \alpha_n, \beta_n, ensuring real, zeros interlacing those of adjacent polynomials. Computational aspects of Laurent polynomials involve Gröbner bases for ideals in multivariate Laurent rings K[X^{\pm 1}_1, \dots, X^{\pm 1}_n], extended via generalized orders on the of exponents. These orders decompose the into cones without invertible elements, enabling a that reduces S-pairs across cones to compute bases, with leading terms defined per cone. This framework solves systems of Laurent equations, as in or , and is implemented in for effective computation.