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Reciprocal polynomial

In , a reciprocal polynomial (also known as a palindromic or anti-palindromic ) is a p(x) of degree n over the real or complex numbers such that p(x) = c x^n p(1/x), where c = \pm 1. If c = 1, it is called palindromic or self-reciprocal; if c = -1, it is anti-palindromic or skew-reciprocal. This condition implies that the coefficients of p(x) = \sum_{k=0}^n a_k x^k satisfy a_k = c a_{n-k} for all k, creating a symmetric (or anti-symmetric) structure in the sequence. A fundamental property of reciprocal polynomials is the symmetry of their roots: if r is a root (with r \neq 0), then $1/r is also a root with the same multiplicity, leading to roots that lie on the unit circle or in reciprocal pairs symmetric with respect to it. This root reciprocity arises naturally in contexts like the eigenvalues of certain matrices, where if \lambda is an eigenvalue, so is $1/\lambda. Reciprocal polynomials frequently appear in linear algebra as the characteristic polynomials of orthogonal, unitary, or matrices, whose spectra exhibit the required reciprocity. They also play roles in and , such as in studying the irreducibility of polynomials over finite fields or the distribution of Galois groups for random reciprocal polynomials. Applications extend to , where self-reciprocal polynomials relate to cyclic codes and their duals, and to combinatorial problems involving unimodal sequences or Mahler measures.

Definition and Basic Properties

Formal Definition

A polynomial p(x) of degree n over a F is called if it satisfies x^n p(1/x) = p(x). It is antipalindromic if x^n p(1/x) = -p(x), with the latter requiring that the of F is not 2 to distinguish the cases. In general form, such a is expressed as p(x) = \sum_{k=0}^n a_k x^k, where the satisfy a_{n-k} = a_k for the case and a_{n-k} = -a_k for the antipalindromic case (again assuming not 2). These conditions follow directly from substituting into the : x^n p(1/x) = \sum_{k=0}^n a_k x^{n-k} = \sum_{j=0}^n a_{n-j} x^j, which equals p(x) or -p(x) under the respective definitions. Reciprocal polynomials are often normalized to be monic, with leading a_n = [1](/page/1). Scalar multiples preserve the or antipalindromic , as x^n (c p(1/x)) = c (x^n p(1/x)) = c p(x) or -c p(x). While polynomials have that come in pairs r, 1/r, the term " equation" refers to polynomial equations where satisfy this pairing without necessarily imposing the coefficient symmetry on the defining .

Coefficient Conditions

A reciprocal polynomial p(x) = \sum_{k=0}^n a_k x^k of n satisfies the x^n p(1/x) = p(x). Expanding the left side yields \sum_{k=0}^n a_k x^{n-k} = \sum_{k=0}^n a_k x^k, which implies the necessary and sufficient condition a_k = a_{n-k} for all k = 0, 1, \dots, n. This symmetry in the coefficients characterizes palindromic reciprocal polynomials. For antipalindromic reciprocal polynomials, the equation becomes x^n p(1/x) = -p(x), leading to the condition a_k = -a_{n-k} for all k. In both cases, the relations pair the coefficients symmetrically from the ends of the sequence. The handling of the middle coefficient depends on the parity of the degree n. For palindromic polynomials, if n = 2m is even, the middle coefficient a_m is arbitrary, while the pairs (a_k, a_{n-k}) for k = 0 to m-1 are equal. If n = 2m+1 is odd, the coefficients a_m and a_{m+1} must satisfy a_m = a_{m+1}, with no further restriction, and the polynomial has a root at x = -1. For antipalindromic polynomials, if n = 2m is even, the middle coefficient satisfies a_m = -a_m, implying a_m = 0, and the polynomial has roots at x = 1 and x = -1. If n = 2m+1 is odd, the middle pair satisfies a_m = -a_{m+1}, with no coefficient forced to zero. Antipalindromic polynomials always have a root at x = 1. Any palindromic reciprocal polynomial can be expressed in the reduced form p(x) = \sum_{k=0}^m a_k (x^k + x^{n-k}) (adjusted for the middle term when n is even by including a_m x^m separately), where the a_k for k = 0 to m are the coefficients. This representation highlights the and confirms that there are exactly m+1 independent coefficients, regardless of whether n = 2m or n = 2m+1. A similar form holds for antipalindromic polynomials, replacing the plus sign with minus, and omitting the middle term when n is even. Thus, any reciprocal polynomial (palindromic or antipalindromic) can be written uniquely in such a with the specified number of coefficients.

Reciprocal Equation

A polynomial satisfies a specific condition derived from the . For a p(x) = \sum_{i=0}^n a_i x^i of degree n, the defines q(x) = x^n p(1/x). The p(x) is if q(x) = \pm p(x), where the positive case corresponds to palindromic polynomials and the negative case to antipalindromic polynomials. To compute q(x), substitute $1/x into p(x) and multiply by x^n: p\left( \frac{1}{x} \right) = \sum_{i=0}^n a_i x^{-i}, \quad q(x) = x^n \sum_{i=0}^n a_i x^{-i} = \sum_{i=0}^n a_i x^{n-i} = \sum_{i=0}^n a_{n-i} x^i. This process reverses the order of the coefficients while preserving the . The resulting q(x) is known as the reversal polynomial of p(x), directly embodying the coefficient reversal. polynomials can be generated by specifying the first \lfloor n/2 \rfloor + 1 coefficients arbitrarily (ensuring the leading coefficient is nonzero) and extending the remaining coefficients symmetrically to satisfy a_i = a_{n-i} or a_i = -a_{n-i}, depending on the sign in the condition. For example, for 4 and the positive case, choosing a_0, a_1, a_2 determines a_3 = a_1 and a_4 = a_0. This parameterization reduces the free parameters to roughly half the .

Types of Reciprocal Polynomials

Palindromic Polynomials

Palindromic polynomials represent the case of reciprocal polynomials where the transformation yields the polynomial itself, satisfying x^n p(1/x) = p(x) for a polynomial p(x) of degree n. This condition is equivalent to the coefficients a_k obeying a_k = a_{n-k} for k = 0, 1, \dots, n. They are also known as self-reciprocal polynomials due to this symmetry. The terminology "palindromic" originates from the property that the sequence of coefficients forms a palindrome, reading identically forwards and backwards. Constant polynomials of degree 0 are trivially palindromic, as they have a single with no reversal to compare. Linear polynomials are palindromic only if they take the form c(x + 1) for some constant c, ensuring the coefficients c, c are symmetric. Illustrative examples include the p(x) = x^2 + 2x + 1 = (x+1)^2, with coefficients 1, 2, 1, and the cubic p(x) = x^3 + 2x^2 + 2x + 1, with coefficients 1, 2, 2, 1; both satisfy the defining equation directly. Many cyclotomic polynomials, such as \Phi_3(x) = x^2 + x + 1, also exhibit this palindromic structure. Palindromic polynomials can be constructed systematically from lower-degree forms by enforcing coefficient symmetry. For degree 2, they are generated as a(x^2 + 1) + b x for constants a, b. For degree 4, the general form is a(x^4 + 1) + b(x^3 + x) + c x^2, where a, b, c are arbitrary constants, ensuring the required pairing of coefficients.

Antipalindromic Polynomials

Antipalindromic polynomials represent the antisymmetric case of polynomials, satisfying the equation x^n p(1/x) = -p(x), where n is the of p(x) = \sum_{k=0}^n a_k x^k. This condition is equivalent to the symmetry a_k = -a_{n-k} for all k = 0, 1, \dots, n. In contrast to palindromic polynomials, which exhibit symmetric a_k = a_{n-k}, the negative sign enforces an antisymmetric structure. The construction of an antipalindromic polynomial involves selecting the coefficients a_0, a_1, \dots, a_{\lfloor n/2 \rfloor} (with a_{n/2} = 0 if n is even, to satisfy the self-paired condition a_{n/2} = -a_{n/2}), and then setting the remaining coefficients as a_{n-k} = -a_k. This ensures the polynomial is fully determined by roughly half its coefficients. Antipalindromic polynomials exist for both even and odd degrees over fields of characteristic not equal to 2; in characteristic 2, the negation becomes trivial, collapsing the definition to the palindromic case. They always vanish at x = 1, since the coefficient sum is zero under the pairing. More precisely, an antipalindromic polynomial of odd degree is a multiple of x - 1 with the quotient being palindromic, while one of even degree is a multiple of x^2 - 1 with the quotient palindromic. Representative examples illustrate these properties. For degree (), p(x) = x - [1](/page/1) has coefficients a_0 = -[1](/page/1), a_1 = [1](/page/1), satisfying a_0 = -a_1, and x p(1/x) = x(1/x - [1](/page/1)) = 1 - x = -(x - [1](/page/1)). For degree (), p(x) = (x - [1](/page/1))^3 = x^3 - [3](/page/3)x^2 + [3](/page/3)x - [1](/page/1) has coefficients -[1](/page/1), [3](/page/3), -[3](/page/3), [1](/page/1), where each pairs antisymmetrically, and dividing by x - [1](/page/1) yields the palindromic x^2 - 2x + [1](/page/1). For degree 2 (even), p(x) = x^2 - [1](/page/1) has coefficients -[1](/page/1), 0, [1](/page/1), with the middle zero, and x^2 p(1/x) = x^2 (1/x^2 - [1](/page/1)) = 1 - x^2 = -(x^2 - [1](/page/1)); dividing by x^2 - [1](/page/1) gives the constant , which is palindromic. These examples highlight how antipalindromic polynomials into a linear or at the reciprocal points and a palindromic remainder, distinguishing them from their palindromic counterparts.

Properties Over Specific Fields

Real Coefficients

For reciprocal polynomials with real coefficients, the roots satisfy a symmetry where if r is a root, then $1/r is also a root, as the reciprocal property p(x) = x^n p(1/x) implies this pairing. Since the coefficients are real, the roots are either real or come in complex conjugate pairs. For real roots r \neq 0, the pair r and $1/r must both be real, so either r = \pm 1 (fixed points under reciprocation) or they form distinct real pairs with r \neq 1/r. This structure ensures that real roots away from \pm 1 appear in reciprocal pairs, influencing the polynomial's factorization over the reals. A standard method to analyze and solve these polynomials exploits the substitution y = x + \frac{1}{x}, which leverages the reciprocity to reduce the equation's degree. For a palindromic p(x) of degree n (where coefficients satisfy a_i = a_{n-i}), divide p(x) by x^{n/2} (valid since x = 0 is not a ), yielding an of the form q(y) = 0, where q is a in y of degree \lfloor n/2 \rfloor. This transformation expresses higher powers of x in terms of y, as relations like x^2 + \frac{1}{x^2} = y^2 - 2 allow recursive reduction, effectively halving the problem's complexity over the reals. The resulting q(y) often exhibits even in certain contexts, facilitating further analysis. For example, consider the palindromic p(x) = x^2 + a x + 1 = 0 with real a. Dividing by x gives x + a + \frac{1}{x} = 0, or y + a = 0, so y = -a. Solving x + \frac{1}{x} = -a then yields the original roots via the , demonstrating the reduction to a in y. This method extends to higher even degrees, where the substituted becomes solvable by techniques once reduced. Regarding non-real roots, complex roots occur in pairs r, 1/r, but with real coefficients, the set is closed under complex conjugation, so each is paired with $1/\bar{r} as well, ensuring quadruple symmetries when necessary (e.g., r, \bar{r}, 1/r, 1/\bar{r}) unless on the unit circle. This closure maintains the polynomial's real coefficient structure while highlighting the interplay between reciprocity and conjugation over the .

Complex Coefficients

For reciprocal polynomials with complex coefficients, the condition p(x) = c x^n p(1/x) with c = \pm 1 implies that the coefficients satisfy a_k = c a_{n-k} for numbers a_k. The are closed under the transformation r \mapsto 1/r (with matching multiplicities), so nonzero roots appear in reciprocal pairs. Unlike polynomials with real coefficients, there is no requirement for roots to come in pairs, allowing greater flexibility in root locations as long as the reciprocal symmetry holds. A notable is that if all of a reciprocal lie on the , then the coefficients must be real (up to by a scalar of 1). This follows because on the unit satisfy $1/r = \bar{r}, imposing conjugate symmetry that forces real coefficients. For example, the palindromic p(x) = x^2 + i x + 1 satisfies a_0 = 1 = a_2, a_1 = i = a_1, and x^2 p(1/x) = p(x), confirming the reciprocal with c = 1. The substitution y = x + 1/x can still be applied to palindromic cases (c=1) to reduce the degree, though the resulting equation in y has complex coefficients and may not simplify as neatly as in the real case.

Advanced Topics

Root Structures

A fundamental property of reciprocal polynomials is that their non-zero roots occur in reciprocal pairs: if r is a root, then $1/r is also a root. This follows directly from the defining relation p(x) = c x^n p(1/x) with c = \pm 1 for a polynomial p(x) of degree n, which implies p(1/r) = 0 whenever p(r) = 0 and r \neq 0. The coefficient symmetry of reciprocal polynomials enforces this root pairing, preserving the algebraic structure across inversion. The multiplicities of these paired roots are equal. If r has multiplicity m, then $1/r also has multiplicity m, as the transformation x \to 1/x is a on the multiset of roots excluding zero, and the polynomial's symmetry maintains the order of vanishing at paired points. Reciprocal polynomials have no root at zero unless the constant term vanishes, in which case the polynomial is divisible by x; however, monic reciprocal polynomials of degree greater than zero have constant term equal to ±1, depending on whether the polynomial is palindromic (c=1) or anti-palindromic (c=-1), excluding zero as a root. When the coefficients are real, the root structure combines reciprocal pairing with complex conjugation. Real roots must occur at \pm [1](/page/1) (where r = 1/r or r = -[1](/page/−1)/r holds self-consistently) or in pairs (r, 1/r) with r real and |r| \neq 1; positive real pairs satisfy r > 0 and $1/r > 0, while negative pairs satisfy r < 0 and $1/r < 0. Non-real roots appear in quartets (r, 1/r, \bar{r}, 1/\bar{r}) unless they lie on the unit circle, where conjugation aligns with inversion ($1/\bar{r} = r if |r| = 1), reducing to conjugate pairs. Self-reciprocal polynomials with real coefficients can be expressed in the form p(z) = z^n q(z + 1/z) for even degree $2n, where q is a polynomial of degree n with real coefficients, further highlighting the paired real roots or unit circle placements. A notable special case involves polynomials with all roots on the unit circle |z| = 1, which are self-inversive (a broader class including reciprocal polynomials with real coefficients satisfying p(z) = z^n p(1/z)). Such polynomials have roots closed under inversion and conjugation, ensuring the entire root set lies on the unit circle; examples include cyclotomic polynomials \Phi_n(x) for n > 1, whose roots are primitive n-th roots of unity, all on the unit circle, and which are themselves reciprocal. More generally, self-inversive polynomials with suitable coefficient conditions (e.g., dominant middle coefficients) guarantee a fixed number of simple roots on the unit circle, such as exactly n - 2l simple unit circle roots under inequalities on the coefficients |a_{n-l}| > \frac{1}{2} \left( \frac{n}{n-2l} \right) \sum_{k \neq l, n-l} |a_k| for l < n/2.

Galois Theory Aspects

The of a reciprocal polynomial of even $2n over \mathbb{Q} acts on the roots while preserving the inherent pairing r \leftrightarrow 1/r, which imposes a specific structure on the group. This action ensures that the Galois group embeds as a of the \mathcal{C}_2 \wr S_n, where \mathcal{C}_2 is the of order 2, reflecting the symmetry between reciprocal root pairs. For certain reciprocal polynomials, such as those of 6, the Galois group can be characterized as a transitive of S_6 isomorphic to dihedral groups or other specific forms, determined through resolvent computations and cycle type analysis. Criteria for the irreducibility of monic reciprocal polynomials over \mathbb{Q} often leverage their palindromic structure, with tests analogous to those for general polynomials but adapted to the reciprocity condition. For instance, a reciprocal polynomial f(x) of degree $2n is irreducible over \mathbb{Q} if it shares no common roots with its reciprocal counterpart in a transformed variable, as established by variants of Eisenstein's criterion tailored to reciprocal forms. Moreover, almost all integer coefficient reciprocal polynomials are irreducible over \mathbb{Q}, with the proportion approaching 1 as the height (maximum absolute coefficient) increases, due to the rarity of factorizations preserving the reciprocal symmetry. These criteria facilitate explicit constructions and probabilistic estimates for irreducibility. Recent developments as of have advanced the understanding of random reciprocal polynomials through probabilistic . For a random monic reciprocal polynomial of degree $2m with coefficients drawn uniformly from [-H, H], the probability of irreducibility over \mathbb{Q} tends to 1 as m \to \infty and H grows appropriately, with the achieving the full hyperoctahedral structure \mathcal{C}_2 \wr S_m with probability approaching 1. This result, building on earlier work, rules out smaller like the maximal alternating subgroup via of cycle structures and resolvents. Related studies confirm that for large degrees, the deviates from the full only with probability O(1/H), primarily due to rational . Generalizations to self-conjugate reciprocal polynomials extend these insights to finite fields, where they play a key role in constructing polynomials. A self-conjugate polynomial over \mathbb{F}_q permutes \mathbb{F}_{q^2} it has no roots on the unit circle in the , enabling explicit families of degree-2 and degree-3 permutation polynomials via linearizations of rational functions. These polynomials inherit irreducibility properties from their base forms and are classified by orders, facilitating applications in and over finite fields.

Applications

Coding Theory

In , reciprocal polynomials, particularly self-reciprocal ones, play a key role in constructing reversible cyclic codes. A cyclic code over a is reversible if the reversal of any codeword is also a codeword, and this property holds the is self-reciprocal, meaning it equals its reciprocal g(x) = x^{\deg g} g(x^{-1}). This symmetry allows for efficient encoding and decoding algorithms that exploit the code's invariance under reversal, originally introduced by Massey in to leverage structural symmetries in error-correcting codes. In BCH codes, self-reciprocal polynomials facilitate the construction of reversible variants, where the is the of self-reciprocal minimal polynomials corresponding to consecutive powers of a . For instance, primitive narrow-sense BCH codes of designed \delta = 2t+1 can be made reversible by selecting cyclotomic factors that are self-reciprocal, ensuring the code parameters remain optimal while adding the reversibility property; an example is the binary BCH code of length 31, dimension 20, and minimum 6 over \mathbb{F}_2. This connection arises because the roots of the in the extension come in reciprocal pairs \beta^i and \beta^{-i}, mirroring the root structures of polynomials over finite fields. Reciprocal polynomials also appear in applications to convolutional codes and Reed-Solomon codes. For convolutional codes, reversibility similarly requires self-reciprocal polynomials, enabling symmetric trellis structures for Viterbi decoding. In Reed-Solomon codes, which are a subclass of BCH codes, the code's is the reciprocal of the original check polynomial, pairing roots \alpha^j with \alpha^{-j} to ensure the code and its share compatible structures for applications like decoding. An illustrative example is a palindromic (self-reciprocal) polynomial for a cyclic , such as g(x) = x^3 + x^2 + x + 1 over \mathbb{F}_2, which generates a self-orthogonal where every codeword is orthogonal to itself under the standard inner product, enhancing properties like minimum distance bounds. The exploitation of reciprocal polynomials in these contexts dates to the early 1960s, with Massey's work emphasizing their utility in simplifying code analysis and implementation for practical correction in data transmission and systems.

and

Self-conjugate reciprocal polynomials play a significant role in by enabling the construction of polynomials over finite fields, which are crucial for designing substitution boxes (S-boxes) in block ciphers and linear feedback shift registers in stream ciphers. In , Zieve introduced a to generate such permutations using bijective Redei functions on subgroups of the of finite fields, where the associated self-conjugate polynomials ensure bijectivity through their root properties outside the unit . Extensions of this work in 2024 have provided explicit constructions of infinite families of permutation binomials and trinomials over quadratic extensions of finite fields, leveraging the absence of on the unit in the self-conjugate form to guarantee permutativity. In number theory, factoring reciprocal polynomials over the rationals benefits from their inherent symmetry, as the reciprocal condition implies that irreducible factors must also exhibit reciprocal or anti-reciprocal properties, simplifying decomposition algorithms. Effective irreducibility criteria for these polynomials, based on evaluating substitutions and resultant computations, have been developed to determine when a reciprocal polynomial remains irreducible over \mathbb{Q}. For instance, irreducible reciprocal polynomials over \mathbb{Q}, such as certain low-degree examples like x^4 + x^3 + x^2 + x + 1, are employed in secure key generation for number-theoretic cryptographic protocols, where their irreducibility ensures the hardness of root-finding problems. The symmetric zero structure of reciprocal polynomials connects to through their role in L-functions, where paired reflect the functional equation's reciprocity, influencing evaluations at critical points related to class number problems in imaginary fields. Recent 2025 investigations into the Galois groups of random reciprocal polynomials of large degree demonstrate that these groups are the full with high probability, providing a foundation for cryptographic hardness assumptions in protocols relying on the intractability of Galois-theoretic computations. In , the self-reciprocal structure of cyclotomic polynomials underpins trapdoor generation in ring-learning with errors (Ring-LWE) schemes, allowing efficient sampling of short vectors via the inherent symmetry that facilitates fast Fourier transform-based operations for and homomorphic operations. This reciprocity enables trapdoors that exploit the ring's , enhancing security against attacks while maintaining computational efficiency.