Polynomials with Symmetric Zeros

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 selfreciprocal 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.

How to count the number of zeros that a polynomial has on the unit circle?

“…Self-inversive polynomials are important in both pure and applied mathematics: they appear in connection with the theory of numbers, algebraic curves, knots theory, error-correcting codes, cryptography and also in some topics of physics as in quantum and statistical mechanics -see [30][31][32][33][34][35]. There is a countless number of papers that presents conditions for all, some, or no zero of a self-inversive polynomial to lie on the unit circle -see [36] and references therein.…”

Orthogonal polynomials and Möbius transformations

“…The orthogonality relation (9) for the Möbius-transformed rational functions R n (x) follows from the change of variable x = M (y) in the integral (5), taking into account that the Jacobian is M ′ (x) = ∆/ (cx + d) 2 . The corresponding orthogonality condition (11) for the Möbius-transformed polynomials Q n (x) follows from their definitions given in (7) and from (9), after we absorb the factors (cx + d) m and (cx + d) n into the varying weight function ω m,n (x). Besides, notice that the path of integration I = {x ∈ R : l < x < r} is mapped through the inverse Möbius-transformation, W (x), onto the curve Γ , as defined in (12), as well as the limits x = l and x = r are respectively mapped to λ = W (l) and ρ = W (r).…”

“…where F (x) and G(x) are the same coefficients that appear in the differential equation (22). Similarly, the degree-dependent weight function ω m,n (x) given in (11), that is associated with the sequence…”

“…where F (x) = F (x) is the same function defined in (24), ω(x) and ω n,n (x) are the new weight functions defined in (10) and (11), respectively, and we introduced the differential operator,…”

“…Consequently, we shall get an infinite sequence of self-inversive or self-reciprocal polynomials (which are complex or real polynomials whose zeros are all symmetric with respect to the unit circle). It is worth to mention that self-inversive and self-reciprocal polynomials are important in many fields of Physics and Mathematics -see [11] and references therein. Finally, such Möbius-transformed polynomials are also orthogonal on the unit circle with respect to a varying measure; because of this, we can conclude that they are intimately related with the usual orthogonal polynomials on the unit circle and also with para-orthogonal polynomials.…”