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In algebra, given a polynomial
with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial, [1] [2] denoted by p∗ or pR, [2] [1] is the polynomial [3]
That is, the coefficients of p∗ are the coefficients of p in reverse order. Reciprocal polynomials arise naturally in linear algebra as the characteristic polynomial of the inverse of a matrix.
In the special case where the field is the complex numbers, when
the conjugate reciprocal polynomial, denoted p†, is defined by,
where denotes the complex conjugate of , and is also called the reciprocal polynomial when no confusion can arise.
A polynomial p is called self-reciprocal or palindromic if p(x) = p∗(x). The coefficients of a self-reciprocal polynomial satisfy ai = an−i for all i.
Reciprocal polynomials have several connections with their original polynomials, including:
Other properties of reciprocal polynomials may be obtained, for instance:
A self-reciprocal polynomial is also called palindromic because its coefficients, when the polynomial is written in the order of ascending or descending powers, form a palindrome. That is, if
is a polynomial of degree n, then P is palindromic if ai = an−i for i = 0, 1, ..., n.
Similarly, a polynomial P of degree n is called antipalindromic if ai = −an−i for i = 0, 1, ..., n. That is, a polynomial P is antipalindromic if P(x) = –P∗(x).
From the properties of the binomial coefficients, it follows that the polynomials P(x) = (x + 1)n are palindromic for all positive integers n, while the polynomials Q(x) = (x – 1)n are palindromic when n is even and antipalindromic when n is odd.
Other examples of palindromic polynomials include cyclotomic polynomials and Eulerian polynomials.
A polynomial with real coefficients all of whose complex roots lie on the unit circle in the complex plane (that is, all the roots have modulus 1) is either palindromic or antipalindromic. [10]
A polynomial is conjugate reciprocal if and self-inversive if for a scale factor ω on the unit circle. [11]
If p(z) is the minimal polynomial of z0 with |z0| = 1, z0 ≠ 1, and p(z) has real coefficients, then p(z) is self-reciprocal. This follows because
So z0 is a root of the polynomial which has degree n. But, the minimal polynomial is unique, hence
for some constant c, i.e. . Sum from i = 0 to n and note that 1 is not a root of p. We conclude that c = 1.
A consequence is that the cyclotomic polynomials Φn are self-reciprocal for n > 1. This is used in the special number field sieve to allow numbers of the form x11 ± 1, x13 ± 1, x15 ± 1 and x21 ± 1 to be factored taking advantage of the algebraic factors by using polynomials of degree 5, 6, 4 and 6 respectively – note that φ ( Euler's totient function) of the exponents are 10, 12, 8 and 12.[ citation needed]
Per Cohn's theorem, a self-inversive polynomial has as many roots in the unit disk as the reciprocal polynomial of its derivative. [12] [13]
The reciprocal polynomial finds a use in the theory of cyclic error correcting codes. Suppose xn − 1 can be factored into the product of two polynomials, say xn − 1 = g(x)p(x). When g(x) generates a cyclic code C, then the reciprocal polynomial p∗ generates C⊥, the orthogonal complement of C. [14] Also, C is self-orthogonal (that is, C ⊆ C⊥), if and only if p∗ divides g(x). [15]
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This article needs additional citations for
verification. (April 2021) |
In algebra, given a polynomial
with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial, [1] [2] denoted by p∗ or pR, [2] [1] is the polynomial [3]
That is, the coefficients of p∗ are the coefficients of p in reverse order. Reciprocal polynomials arise naturally in linear algebra as the characteristic polynomial of the inverse of a matrix.
In the special case where the field is the complex numbers, when
the conjugate reciprocal polynomial, denoted p†, is defined by,
where denotes the complex conjugate of , and is also called the reciprocal polynomial when no confusion can arise.
A polynomial p is called self-reciprocal or palindromic if p(x) = p∗(x). The coefficients of a self-reciprocal polynomial satisfy ai = an−i for all i.
Reciprocal polynomials have several connections with their original polynomials, including:
Other properties of reciprocal polynomials may be obtained, for instance:
A self-reciprocal polynomial is also called palindromic because its coefficients, when the polynomial is written in the order of ascending or descending powers, form a palindrome. That is, if
is a polynomial of degree n, then P is palindromic if ai = an−i for i = 0, 1, ..., n.
Similarly, a polynomial P of degree n is called antipalindromic if ai = −an−i for i = 0, 1, ..., n. That is, a polynomial P is antipalindromic if P(x) = –P∗(x).
From the properties of the binomial coefficients, it follows that the polynomials P(x) = (x + 1)n are palindromic for all positive integers n, while the polynomials Q(x) = (x – 1)n are palindromic when n is even and antipalindromic when n is odd.
Other examples of palindromic polynomials include cyclotomic polynomials and Eulerian polynomials.
A polynomial with real coefficients all of whose complex roots lie on the unit circle in the complex plane (that is, all the roots have modulus 1) is either palindromic or antipalindromic. [10]
A polynomial is conjugate reciprocal if and self-inversive if for a scale factor ω on the unit circle. [11]
If p(z) is the minimal polynomial of z0 with |z0| = 1, z0 ≠ 1, and p(z) has real coefficients, then p(z) is self-reciprocal. This follows because
So z0 is a root of the polynomial which has degree n. But, the minimal polynomial is unique, hence
for some constant c, i.e. . Sum from i = 0 to n and note that 1 is not a root of p. We conclude that c = 1.
A consequence is that the cyclotomic polynomials Φn are self-reciprocal for n > 1. This is used in the special number field sieve to allow numbers of the form x11 ± 1, x13 ± 1, x15 ± 1 and x21 ± 1 to be factored taking advantage of the algebraic factors by using polynomials of degree 5, 6, 4 and 6 respectively – note that φ ( Euler's totient function) of the exponents are 10, 12, 8 and 12.[ citation needed]
Per Cohn's theorem, a self-inversive polynomial has as many roots in the unit disk as the reciprocal polynomial of its derivative. [12] [13]
The reciprocal polynomial finds a use in the theory of cyclic error correcting codes. Suppose xn − 1 can be factored into the product of two polynomials, say xn − 1 = g(x)p(x). When g(x) generates a cyclic code C, then the reciprocal polynomial p∗ generates C⊥, the orthogonal complement of C. [14] Also, C is self-orthogonal (that is, C ⊆ C⊥), if and only if p∗ divides g(x). [15]
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