This paper establishes conditions guaranteeing the reality of all the zeros of polynomials Pn(z) in the polynomial sequence {Pn(z)}n=1∞ satisfying a five-term recurrence relation Pn(z) = zPn−1(z) + αPn−2(z) + βPn−3(z) + γPn−4(z) with the standard initial conditions P(z) = 1 , P− 1(z) = P− 2(z) = P− 3(z) = 0 , where α, β, γ are real coefficients, γ ≠ 0 and z is a complex variable. We interpret this sequence of polynomials as principal minors of an appropriate banded Toeplitz matrix whose associated Laurent polynomial b(z) is holomorphic in ℂ {0}. We derive a criterion for the preimage b−1 (ℝ) to contain a Jordan curve, which, by a recent result of Shapiro and Stampach, is necessary and sufficient for every polynomial in the sequence {Pn(z)}n=1∞ to be hyperbolic.

This paper discusses the location of zeros of polynomials in a polynomial sequence {Pn(z)}∞n=1 generated by a three-term recurrence relation of the form Pn(z)+B(z)Pn−1(z)+A(z)Pn−k(z)=0 with k>2 and the standard initial conditions P0(z)=1, P−1(z)=P−k+1(z)=0, where A(z) and B(z) are arbitrary coprime real polynomials. We show that there always exist polynomials in {Pn(z)}∞n=1 with nonreal zeros.

In this thesis, we study the problem of location of the zeros of individual polynomials in sequences of polynomials generated by linear recurrence relations.

In paper I, we establish the necessary and sufficient conditions that guarantee hyperbolicity of all the polynomials generated by a three-term recurrence of length 2, whose coefficients are arbitrary real polynomials. These zeros are dense on the real intervals of an explicitly defined real semialgebraic curve.

Paper II extends Paper I to three-term recurrences of length greater than 2. We prove that there always exist non-hyperbolic polynomial(s) in the generated sequence. We further show that with at most finitely many known exceptions, all the zeros of all the polynomials generated by the recurrence lie and are dense on an explicitly defined real semialgebraic curve which consists of real intervals and non-real segments. The boundary points of this curve form a subset of zero locus of the discriminant of the characteristic polynomial of the recurrence.

Paper III discusses the zero set for polynomials generated by three-term recurrences of lengths 3 and 4 with arbitrary polynomial coefficients. We prove that except the zeros of the polynomial coefficients, all the zeros of every generated polynomial lie on an explicitly defined real semialgebraic curve.

Paper IV extends the results in paper III and generalizes a conjecture by K. Tran [2]. We consider a three-term recurrence relation of any length whose coefficients are arbitrary complex polynomials and prove that with the exception of the zeros of the polynomial coefficients, all the zeros of every generated polynomial lie on a real algebraic curve. We derive the equation of this curve.

Paper V establishes the necessary and sufficient conditions guaranteeing the reality of all the zeros of every polynomial generated by a special five-term recurrence with real coefficients. We put the problem in the context of banded Toeplitz matrices whose associated Laurent polynomial is holomorphic in the punctured plane. We interpret the conditions in terms of the positivity/negativity of the discriminant of a certain polynomial whose coefficients are explicit functions of the parameters in the recurrence.

This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence {P-i}(i=1)(infinity) generated by a three-term recurrence relation P-i(x) + Q(1)(x)P-i (1)(x) + Q(2)(x)Pi-2(x) = 0 with the standard initial conditions P-0(x) = 1, P-1(x) = 0, where Q(1)(x) and Q(2)(x) are arbitrary real polynomials.

A conjecture of Khang Tran  claims that for an arbitrary pair of polynomials A(z) and B(z), every zero of every polynomial in the sequence {P_n(z)} satisfying the three-term recurrence relation of length k

P_n(z) + B(z)P_{n−1}(z) + A(z)P_{n−k}(z) = 0

with the standard initial conditions P_0(z) = 1, P_{−1}(z) = · · · = P_{−k+1}(z) = 0 which is not a zero of A(z) lies on the real (semi)-algebraic curve C  given by

Im(( B^k(z)/ A(z)) = 0 and 0 ≤ (−1)^k ≤ Re(( B^k(z)/ A(z)) ≤ k^k (k − 1)^{k−1}. In this short note, we show that for the recurrence relation (generalizing the latter recurrence of Tran) given by

P_n(z) + B(z)P_{n−l}(z) + A(z)P_{n−k}(z) = 0, with coprime k and l and the same standard initial conditions as above, every root of P_n(z) which is not a zero of A(z)B(z) belongs to the real algebraic curve C_{l,k} given by

Im(( B^k(z)/ A(z)) = 0.

This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence {Pi } generated by a three-term recurrence relation P_i (x) + Q_1 (x)P_{i−1} (x) + Q_2 (x)P_{i−2}(x) = 0 with the standard initial conditions P_0(x) = 1, P_{-1} (x) = 0, where Q_1(x) and Q_2(x) are arbitrary real polynomials.

This paper discusses the location of zeros of polynomials in a polynomial sequence {P_n(z)} generated by a three-term recurrence relation of the form P_n(z)+B(z)P_{n−1}(z)+A(z)P_{n−k}(z)=0 with k>2 and the standard initial conditions P_0(z)=1,P_{−1}(z)=…=P_{−k+1}(z)=0, where A(z) and B(z) are arbitrary coprime real polynomials. We show that there always exist polynomials in {Pn(z)} with non-real zeros.

Below we establish the conditions guaranteeing the reality of all the zeros of polynomials P_n(z) in the polynomial sequence {P_n(z)} satisfying a five-term recurrence relation

P_n(z) = zP_{n−1}(z) + αP_{n−2}(z) + βP_{n−3}(z) + γP_{n−4}(z),

with the standard initial conditions P_0(z) = 1, P_{−1}(z) = P_{−2}(z) = P_{−3}(z) = 0, where α, β, γ are real coefficients, γ ≠ 0 and z is a complex variable. We interprete this sequence of polynomials as principal minors of an appropriate banded Teoplitz matrix whose associated Laurent polynomial b(z) is holomorphic in C \ {0}. We show that when either all the critical points in the complex plane of b(z) are real; or only two are real together with a pair of complex conjugate critical points and some extra conditions on the parameters, the set b^{−1}(R) contains a Jordan curve with 0 in its interior and in some cases a nonsimple curve enclosing 0. The presence of the said curves is necessary and sufficient for every polynomial in the sequence {P_n(z)} to be hyperbolic.