Alexander polynomials for prime knots up to six crossings.

## Abstract

Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to the real line; the self-inversive polynomials, whose zeros are symmetric to the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric by an inversion with respect to the unit circle followed by a reflection in the real line. Real self-reciprocal polynomials are simultaneously self-conjugate and self-inversive so that their zeros are symmetric to both the real line and the unit circle. In this survey, we present a short review of these polynomials, focusing on the distribution of their zeros.

### Keywords

- self-inversive polynomials
- self-reciprocal polynomials
- Pisot and Salem polynomials
- Möbius transformations
- knot theory
- Bethe equations

## 1. Introduction

In this work, we consider the theory of self-conjugate (SC), self-reciprocal (SR), and self-inversive (SI) polynomials. These are polynomials whose zeros are symmetric either to the real line

## 2. Self-conjugate, self-reciprocal, and self-inversive polynomials

We begin with some definitions:

**Definition 1.** Let *conjugate polynomial* *reciprocal polynomial* *inversive polynomial*

where the bar means complex conjugation. Notice that the conjugate, reciprocal, and inversive polynomials can also be defined without making reference to the coefficients of

From these relations, we plainly see that if

These properties encourage us to introduce the following classes of polynomials:

**Definition 2.** A complex polynomial ^{1} *self-conjugate* (SC), *self-reciprocal* (SR), or *self-inversive* (SI) if, for any zero

Thus, the zeros of any SC polynomial are all symmetric to the real line

**Theorem 1.** *If* *is an SC polynomial of odd degree, then it necessarily has at least one zero on* *. Similarly, if* *is an SR or SI polynomial of odd degree, then it necessarily has at least one zero on*

*Proof.* From Definition 2 it follows that the number of non-real zeros of an SC polynomial

**Theorem 2.** *The necessary and sufficient condition for a complex polynomial* *to be SC, SR, or SI is that there exists a complex number* *of modulus* *so that one of the following relations, respectively, holds:*

*Proof.* It is clear in view of (1) and (2) that these conditions are sufficient. We need to show, therefore, that these conditions are also necessary. Let us suppose first that

with

with

Now from (1), (2) and (3), we can conclude that the coefficients of an SC, an SR, and an SI polynomial of degree

We highlight that any real polynomial is SC—in fact, many theorems which are valid for real polynomials are also valid for, or can be easily extended to, SC polynomials.

There also exist polynomials whose zeros are symmetric with respect to *both* the real line

**Definition 3.** A real self-reciprocal polynomial *positive self-reciprocal* (PSR) polynomial if *negative self-reciprocal* (NSR) polynomial if

Thus, the coefficients of any PSR polynomial

Some elementary properties of PSR and NSR polynomials are the following: first, notice that, if

We also mention that any PSR polynomial of even degree (say,

an expression that is obtained by using the relations

**Theorem 3.** *Let* *be a PSR polynomial of even degree* *. For each pair* *and* *of self-reciprocal zeros of* *that lie on* *, there is a corresponding zero* *of the polynomial* *, as defined above, in the interval* *of the real line.*

*Proof.* For each zero

Finally, remembering that the Chebyshev polynomials of first kind,

## 3. How these polynomials are related to each other?

In this section, we shall analyze how SC, SR, and SI polynomials are related to each other. Let us begin with the relationship between the SR and SI polynomials, which is actually very simple: indeed, from (1), (2), and (3) we can see that each one is nothing but the conjugate polynomial of the other, that is

Thus, if

The relationship between SC and SI polynomials is not so easy to perceive. A way of revealing their connection is to make use of a suitable pair of *Möbius transformations*, that maps the unit circle onto the real line and vice versa, which is often called *Cayley transformations*, defined through the formulas:

This approach was developed in [4], where some algorithms for counting the number of zeros that a complex polynomial has on the unit circle were also formulated.

It is an easy matter to verify that

Given a polynomial *Möbius-transformed polynomials*, namely

The following theorem shows us how the zeros of

**Theorem 4.** *Let* *denote the zeros of* *and* *the respective zeros of* *. Provided* *, we have that* *. Similarly, if* *are the zeros of* *, then we have* *, provided that*

*Proof.* In fact, inverting the expression for

This result also shows that

The following theorem shows that the set of SI polynomials are isomorphic to the set of SC polynomials:

**Theorem 5.** *Let* *be an SI polynomial. Then, the transformed polynomial* *is an SC polynomial. Similarly, if* *is an SC polynomial, then* *will be an SI polynomial.*

*Proof.* Let

Thus, any pair of zeros of

Thus, any pair of zeros of

We can also verify that any SI polynomial with

## 4. Zeros location theorems

In this section, we shall discuss some theorems regarding the distribution of the zeros of SC, SR, and SI polynomials on the complex plane. Some general theorems relying on the number of zeros that an arbitrary complex polynomial has inside, on, or outside

### 4.1 Polynomials that do not necessarily have symmetric zeros

The following theorems are classics (see [1] for the proofs):

**Theorem 6. (Rouché)***. Let* *and* *be polynomials such that* *along all points of* *. Then, the polynomial* *has the same number of zeros inside* *as the polynomial* *, counted with multiplicity.*

Thus, if a complex polynomial

**Theorem 7. (Gauss and Lucas)** *The zeros of the derivative* *of a polynomial* *lie all within the convex hull of the zeros of the*

Thereby, if a polynomial

**Theorem 8. (Cohn)** *A necessary and sufficient condition for all the zeros of a complex polynomial* *to lie on* *is that* *is SI and that its derivative* *does not have any zero outside*

Cohn introduced his theorem in [5]. Bonsall and Marden presented a simpler proof of Conh’s theorem in [6] (see also [7]) and applied it to SI polynomials—in fact, this was probably the first paper to use the expression “self-inversive.” Other important result of Cohn is the following: all the zeros of a complex polynomial

Restricting ourselves to polynomials with real coefficients, Eneström and Kakeya [8, 9, 10] independently presented the following theorem:

**Theorem 9. (Eneström and Kakeya)** *Let* *be a polynomial of degree* *with real coefficients. If its coefficients are such that* *, then all the zeros of* *lie in or on* *. Likewise, if the coefficients of* *are such that* *, then all the zeros of* *lie on or outside*

The following theorems are relatively more recent. The distribution of the zeros of a complex polynomial regarding the unit circle

**Theorem 10. (Marden and Jury)** *Let* *be a complex polynomial of degree* *and* *its reciprocal. Construct the sequence of polynomials* *such that* *and* *for* *so that we have the relations* *. Let* *denote the constant terms of the polynomials* *, i.e.,* *and* *. Thus, if* *of the products* *are negative and* *of the products* *are positive so that none of them are zero, then* *has* *zeros inside* *,* *zeros outside* *and no zero on* *. On the other hand, if* *for some* *but* *, then* *has either* *zeros on* *or* *zeros symmetric to* *. It has additionally* *zeros inside* *and* *zeros outside*

A simple necessary and sufficient condition for all the zeros of a complex polynomial to lie on

**Theorem 11. (Chen)** *A necessary and sufficient condition for all the zeros of a complex polynomial* *of degree* *to lie on* *is that there exists a polynomial* *of degree* *whose zeros are all in or on* *and such that* *for some complex number* *of modulus*

We close this section by mentioning that there exist many other well-known theorems regarding the distribution of the zeros of complex polynomials. We can cite, for example, the famous *rule of Descartes* (the number of positive zeros of a real polynomial is limited from above by the number of sign variations in the ordered sequence of its coefficients), the *Sturm Theorem* (the exact number of zeros that a real polynomial has in a given interval *Kronecker Theorem* (if all the zeros of a monic polynomial with integer coefficients lie on the unit circle, then all these zeros are indeed roots of unity), see [1] for more. There are still other important theorems relying on matrix methods and quadratic forms that were developed by several authors as Cohn, Schur, Hermite, Sylvester, Hurwitz, Krein, among others, see [13].

### 4.2 Real self-reciprocal polynomials

Let us now consider real SR polynomials. The theorems below are usually applied to PSR polynomials, but some of them can be extended to NSR polynomials as well.

An analog of Eneström-Kakeya theorem for PSR polynomials was found by Chen in [12] and then, in a slightly stronger version, by Chinen in [14]:

**Theorem 12. (Chen and Chinen)** *Let* *be a PSR polynomial of degree* *that is written in the form* *and such that* *. Then all the zeros of* *are on*

Going in the same direction, Choo found in [15] the following condition:

**Theorem 13. (Choo)** *Let* *be a PSR polynomial of degree* *and such that its coefficients satisfy the following conditions:* *and* *for* *. Then, all the zeros of* *are on*

Lakatos discussed the separation of the zeros on the unit circle of PSR polynomials in [16]; she also found several sufficient conditions for their zeros to be all on

**Theorem 14. (Lakatos)** *Let* *be a PSR polynomial of degree* *. If* *, then all the zeros of* *lie on* *. Moreover, the zeros of* *are all simple, except when the equality takes place.*

For PSR polynomials of odd degree, Lakatos and Losonczi [17] found a stronger version of this result:

**Theorem 15. (Lakatos and Losonczi)** *Let* *be a PSR polynomial of odd degree, say* *. If* *, where* *then all the zeros of* *lie on* *. The zeros are simple except when the equality is strict.*

Theorem 14 was generalized further by Lakatos and Losonczi in [18]:

**Theorem 16. (Lakatos and Losonczi)** *All zeros of a PSR polynomial* *of degree* *lie on* *if the following conditions hold:* *,* *, and* *, for*

Other conditions for all the zeros of a PSR polynomial to lie on

**Theorem 17. (Kwon)** *Let* *be a PSR polynomial of even degree* *whose leading coefficient* *is positive and* *. In this case, all the zeros of* *will lie on* *if, either* *, or* *and*

Modified forms of this theorem hold for PSR polynomials of odd degree and for the case where the coefficients of

Other interesting results are the following: Konvalina and Matache [21] found conditions under which a PSR polynomial has at least one non-real zero on *Littlewood*, *Newman*, and *Borwein polynomials*—were studied by several authors, see [27, 28, 29, 30, 31, 32, 33, 34, 35] and references therein.^{2} Zeros of the so-called *Ramanujan Polynomials* and generalizations were analyzed in [37, 38, 39]. Finally, the Galois theory of PSR polynomials was studied in [40] by Lindstrøm, who showed that any PSR polynomial of degree less than

### 4.3 Complex self-reciprocal and self-inversive polynomials

Let us consider now the case of complex SR polynomials and SI polynomials. Here, we remark that many of the theorems that hold for SI polynomials either also hold for SR polynomials or can be easily adapted to this case (the opposite is also true).

**Theorem 18. (Cohn)** *An SI polynomial* *has as many zeros outside* *as does its derivative*

This follows directly from Cohn’s Theorem 8 for the case where

O’Hara and Rodriguez [41] showed that the following conditions are always satisfied by SI polynomials whose zeros are all on

**Theorem 19. (O’Hara and Rodriguez)** *Let* *be an SI polynomial of degree* *whose zeros are all on* *. Then, the following inequality holds:* *, where* *denotes the maximum modulus of* *on the unit circle; besides, if this inequality is strict then the zeros of* *are rotations of* *th roots of unity. Moreover, the following inequalities are also satisfied:*

Schinzel in [42], generalized Lakatos Theorem 14 for SI polynomials:

**Theorem 20. (Schinzel)** *Let* *be an SI polynomial of degree* *. If the inequality* *, then all the zeros of* *lie on* *. These zero are simple whenever the equality is strict.*

In a similar way, Losonczi and Schinzel [43] generalized theorem 15 for the SI case:

**Theorem 21. (Losonczi and Schinzel)** *Let* *be an SI polynomial of odd degree, i.e.,* *. If* *, where* *, then all the zeros of* *lie on* *. The zeros are simple except when the equality is strict.*

Another sufficient condition for all the zeros of an SI polynomial to lie on

**Theorem 22. (Lakatos and Losonczi)** *Let* *be an SI polynomial of degree* *and suppose that the inequality* *holds. Then, all the zeros of* *lie on* *. Moreover, the zeros are all simple except when an equality takes place.*

In [45], Lakatos and Losonczi also formulated a theorem that contains as special cases many of the previous results:

**Theorem 23. (Lakatos and Losonczi)** *Let* *be an SI polynomial of degree* *and* *,* *, and* *be complex numbers such that* *,* *, and* *,* *. If* *, then, all the zeros of* *lie on* *. Moreover, these zeros are simple if the inequality is strict.*

In [46], Losonczi presented the following necessary and sufficient conditions for all the zeros of a (complex) SR polynomial of even degree to lie on

**Theorem 24. (Losonczi)** *Let* *be a monic complex SR polynomial of even degree, say* *. Then, all the zeros of* *will lie on* *if, and only if, there exist real numbers* *, all with moduli less than or equal to* *, that satisfy the inequalities:* *,* *, where* *denotes the* *th elementary symmetric function in the* *variables*

Losonczi, in [46], also showed that if all the zeros of a complex monic reciprocal polynomial are on

The theorems above give conditions for *all* the zeros of SI or SR polynomials to lie on *given number* of zeros (or *none*) on the unit circle. Considering this problem, Vieira in [47] found sufficient conditions for an SI polynomial of degree

**Theorem 25. (Vieira)** *Let* *be an SI polynomial of degree* *. If the inequality* *,* *, holds true, then* *will have exactly* *zeros on* *; besides, all these zeros are simple when the inequality is strict. Moreover,* *will have no zero on* *if, for* *even and* *, the inequality* *is satisfied.*

The case

Other results on the distribution of zeros of SI polynomials include the following: Sinclair and Vaaler [49] showed that a monic SI polynomial

## 5. Where these polynomials are found?

In this section, we shall briefly discuss some important or recent applications of the theory of polynomials with symmetric zeros. We remark, however, that our selection is by no means exhaustive: for example, SR and SI polynomials also find applications in many fields of mathematics (e.g., information and coding theory [55], algebraic curves over a finite field and cryptography [56], elliptic functions [57], number theory [58], etc.) and physics (e.g., Lee-Yang theorem in statistical physics [59], Poincaré Polynomials defined on Calabi-Yau manifolds of superstring theory [60], etc.).

### 5.1 Polynomials with small Mahler measure

Given a monic polynomial

where ^{3} of

**Problem 1. (Mahler)** *Find the monic, integer, non-cyclotomic polynomial with the smallest Mahler measure.*

This is an 80-year-old open problem of mathematics. Of course, we can expect that the polynomials with the smallest Mahler measure be among those with only a few number of zeros outside *Pisot polynomial* and its unique zero of modulus greater than *Pisot number* [62]. A breakthrough towards the solution of Mahler’s problem was given by Smyth in [63]:

**Theorem 26. (Smyth)** *The Pisot polynomial* *is the polynomial with smallest Mahler measure among the set of all monic, integer, and non-SR polynomials. Its Mahler measure is given by the value of its Pisot number, which is,*

The Mahler problem is, however, still open for SR polynomials. A monic integer SR polynomial with exactly two (real and positive) zeros (say, *Salem polynomial* [62, 64]. It can be shown that a Pisot polynomial with at least one zero on *Salem number*, which also equals the value of its Mahler measure. A Salem number

**Conjecture 1. (Lehmer)** *The monic integer polynomial with the smallest Mahler measure is the Lehmer polynomial* *a Salem polynomial whose Mahler measure is* *, known as Lehmer’s constant.*

The proof of this conjecture is also an open problem. To be fair, we do not even know if there exists a smallest Salem number at all. This is the content of another problem raised by Lehmer:

**Problem 2. (Lehmer)** *Answer whether there exists or not a positive number* *such that the Mahler measure of any monic, integer, and non-cyclotomic polynomial* *satisfies the inequality*

Lehmer’s polynomial also appears in connection with several fields of mathematics. Many examples are discussed in Hironaka’s paper [68]; here we shall only present an amazing identity found by Bailey and Broadhurst in [69] in their works on polylogarithm ladders: if

### 5.2 Knot theory

A *knot* is a closed, non-intersecting, one-dimensional curve embedded on

One of the most important questions in knot theory is to determine whether or not two knots are equivalent. This, however, is not an easy task. A way of attacking this question is to look for abstract objects—mainly the so-called *knot invariants*—rather than to the knots themselves. A knot invariant is a (topologic, combinatorial, algebraic, etc.) quantity that can be computed for any knot and that is always the same for equivalents knots.^{4} An important class of knot invariants is constituted by the so-called *Knot Polynomials*. Knot polynomials were introduced in 1928 by Alexander [71]. They consist in polynomials with integer coefficients that can be written down for every knot. For about 60 years since its creation, Alexander polynomials were the only known kind of knot polynomial. It was only in 1985 that Jones [72] came up with a new kind of knot polynomials—today known as *Jones polynomials*—and since then other kinds were discovered as well, see [70].

What is interesting for us here is that the Alexander polynomials are PSR polynomials of even degree (say, ^{5} Thus, they have the following general form:

where

Knot | Alexander polynomial | Knot | Alexander polynomial |
---|---|---|---|

Knots theory finds applications in many fields of mathematics in physics—see [70]. In mathematics, we can cite a very interesting connection between Alexander polynomials and the theory of Salem numbers: more precisely, the Alexander polynomial associated with the so-called *Pretzel Knot*

### 5.3 Bethe equations

*Bethe equations* were introduced in 1931 by Hans Bethe [75], together with his powerful method—the so-called *Bethe Ansatz Method*—for solving spectral problems associated with exactly integrable models of statistical mechanics. They consist in a system of coupled and non-linear equations that ensure the consistency of the Bethe Ansatz. In fact, for the XXZ Heisenberg spin chain, the Bethe Equations consist in a coupled system of trigonometric equations; however, after a change of variables is performed, we can write them in the following rational form:

where *L* ∈ *N* ∈ *Δ* ∈

In [76], Vieira and Lima-Santos showed that the solutions of (18), for

Now, from the relation

We can easily verify that the polynomial

as follows: if

Finally, we highlight that the polynomial

### 5.4 Orthogonal polynomials

An infinite sequence *orthogonal polynomial sequence* on the interval

where

Very recently, Vieira and Botta [78, 79] studied the action of Möbius transformations over orthogonal polynomial sequences on the real line. In particular, they showed that the infinite sequence

Hermite polynomials | Möbius-transformed Hermite polynomials |
---|---|

## 6. Conclusions

In this work, we reviewed the theory of self-conjugate, self-reciprocal, and self-inversive polynomials. We discussed their main properties, how they are related to each other, the main theorems regarding the distribution of their zeros and some applications of these polynomials both in physics and mathematics. We hope that this short review suits for a compact introduction of the subject, paving the way for further developments in this interesting field of research.

## Acknowledgments

We thank the editorial staff for all the support during the publishing process and also the *Coordination for the Improvement of Higher Education* (CAPES).

## Notes

- The reader should be aware that there is no standard in naming these polynomials. For instance, what we call here self-inversive polynomials are sometimes called self-reciprocal polynomials. What we mean positive self-reciprocal polynomials are usually just called self-reciprocal or yet palindrome polynomials (because their coefficients are the same whether they are read from forwards or backwards), as well as, negative self-reciprocal polynomials are usually called skew-reciprocal, anti-reciprocal, or yet anti-palindrome polynomials.
- The zeros of such polynomials present a fractal behavior, as was first discovered by Odlyzko and Poonen in [36].
- The Mahler measure of a monic integer polynomial p z can also be defined without making reference to its zeros through the formula M p z = exp ∫ 0 1 log p e 2 πit dt —see [61].
- We remark, however, that different knots can have the same knot invariant. Up to date, we do not know whether there exists a knot invariant that distinguishes all non-equivalent knots from each other (although there do exist some invariants that distinguish every knot from the trivial knot). Thus, until now the concept of knot invariants only partially solves the problem of knot classification.
- Alexander polynomials can also be defined as Laurent polynomials, see [70].