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Roots of Polynomials and Umbilics of Surfaces

For certain polynomials we relate the number of roots inside the unit circle with the index of a non-degenerate isolated umbilic point on a real analytic surface in Euclidean 3-space. In particular, for N>0 we prove that for a certain (N+2)-real dimensional family of complex polynomials of degree N, the number of roots inside the unit circle is less than or equal to 1+N/2. This bound is established as follows. From the polynomial we construct a convex real analytic surface containing an isolated umbilic point, such that the index of the umbilic point is determined by the number of roots of the polynomial that lie inside the unit circle. The bound on the number of roots then follows from Hamburger's bound on the index of an isolated umbilic point on a convex real analytic surface.


Introduction
Almost two centuries ago Cauchy established that all zeros of a complex polynomial must lie inside a circle of a radius that can be determined by the moduli of the polynomial's coefficients [1]. Over the intervening years a vast number of relations have been found between polynomial roots lying inside a circle or annulus, and the coefficients of the polynomial [2] [9] [11].
The determination of the location of the zeros of a given polynomial relative to the unit circle in the complex plane arises in questions of stability in many areas of applied mathematics, including numerical integration, difference equations, control theory and economic growth models [3] [10] [12] [13].
In this paper we present a relationship between the number of zeros contained inside the unit circle of certain polynomials and the index of isolated umbilic points on certain real analytic convex surfaces in R 3 . A deep result of Hamburger [8] bounds the index of an isolated umbilic point on any real analytic convex surface, which yields a bound on the number of zeros inside the unit circle.
Umbilic points have many subtle properties -they can control the evolution of surfaces under curvature flows [7], while Hamburger's index bound does not hold for even small perturbations of the ambient Euclidean metric [4].
The specific polynomials considered in this paper have relations between their coefficients that arise from a Lagrangian condition in the space of oriented affine lines in R 3 .
In more detail, let P N be a polynomial over C of degree N : where the weighted symmetric functions of the roots {ζ n } N n=1 are defined by Consider the following conditions on the coefficients of P N : where N = 2l + 1 or N = 2l + 2. Note that in either case, these are N − 2 real conditions on the coefficients of P N , which give rise to a real N + 2 dimensional family of degree N polynomials. The second of conditions (1) is complex and does not depend solely on the moduli of the polynomial coefficients. Moreover, our approach is complimentary to Cauchy's in that the arguments of the coefficients play a dominant role.
The main result of this paper is: Main Theorem: Let P N be a polynomial satisfying the relations (1) and suppose that none of the roots ζ n lies on the unit circle. Then the number of roots (counted with multiplicity) of P N inside the unit circle is less than or equal to 1 + N/2.
We prove this result by showing in Theorem 3.3 that, given a polynomial P N satisfying conditions (1), there exists a convex real analytic surface in R 3 with an isolated umbilic point which has index I = K − N/2, where K is the number of zeros of P N lying inside the unit circle. Thus, the local umbilic index bound of [8], which states that I ≤ 1, yields the stated result.
In the next section we give the background on the construction of convex surfaces, using Lagrangian surfaces in the space of oriented affine lines in R 3 -further details can be found in [5]. Section 3 contains the proof of the Main Theorem and in the final section we illustrate our techniques by constructing explicit examples.

Real Analytic Convex Surfaces in R 3
Definition 2.1. Let L be the set of oriented (affine) lines in Euclidean 3-space R 3 . Definition 2.2. Let Φ : T S 2 → L be the map that identifies L with the tangent bundle to the unit 2-sphere in Euclidean R 3 , by parallel translation. This bijection gives L the structure of a differentiable 4-manifold [6].
Let (ξ, η) be holomorphic coordinates on T S 2 , where ξ is obtained by stereographic projection from the south pole onto the plane through the equator, and we identify (ξ, η) with the vector

Proposition 2.3. [5]
The map Φ takes (ξ, η)∈ T S 2 to the oriented line given by are Euclidean coordinates on R 3 = C ⊕ R and r is an affine parameter along the line.
An oriented line congruence (or line congruence for short) is a surface Σ ⊂ L. An oriented surface S in R 3 gives rise to a line congruence by its oriented normal lines. Conversely a line congruence is called Lagrangian if it is orthogonal to a (possibly singular) surface in R 3 . The term refers to the canonical symplectic structure on L which vanishes on such a surface -see [6] for further details of the geometric background.
Proposition 2.4. The oriented normals of a convex surface in R 3 form a surface Σ ⊂ L which is the graph of a local section of the bundle π : L → S 2 .
Such congruences are given by η = F (ξ,ξ), where F is a complex-valued function on an open neighbourhood of S 2 . For such local sections, a necessary and sufficient condition for the line congruence to be Lagrangian is: Proposition 2.5. The graph of a local section of π : L → S 2 is Lagrangian iff the defining function F satisfies where, here and throughout, we denote ∂ ∂ξ by ∂. In particular, there exists a real function r : C → R, called the support function, satisfying Given a support function r and the associated F defined by (5), equations (2) and (3) yield a parameterised surface in R 3 . Thus we obtain a surface in 3-space parameterised by the inverse of its Gauss map. Note that r is defined up to an additive constant -replacing r by r + C moves the surface along its normal lines to the parallel surface.
For a convex real analytic surface the graph function F is real analytic and we can expand it in a power series about ξ = 0: Proposition 2.6. The Lagrangian conditions (4) restrict the coefficients of the power series by where n, m ≥ 1.
Proof. Suppose that A nm ξ nξm , nA nm ξ n−1ξm . Then The result then follows from noting that the condition (4) is equivalent to and comparing terms in the above expression.
We now rearrange these conditions into a more convenient form.
Proposition 2.7. The Lagrangian conditions can be rewritten for n > 1, m < n.
From here on, we assume that S is a convex real analytic surface in R 3 and the oriented normals are given by ξ → (ξ, η = F (ξ,ξ)), with power series expansion (6) satisfying I to V of Proposition 2.7.
We are interested in the umbilic points on S, that is, the points where the second fundamental form of S has a double eigenvalue. In our formalism, these points can be characterised as follows: Suppose that S has an isolated umbilic point situated at the point with ξ = 0, and define the polar coordinates (R, θ) by ξ = Re iθ . Definition 2.9. The index I of the isolated umbilic point at 0 is Example 2.10. For the triaxial ellipsoid with semi-axes a 1 , a 2 and a 3 the support function is , and the Lagrangian section is For a 1 = a 2 = a 3 the surface is the rotationally symmetric ellipsoid (the x 3 −axis being the axis of symmetry). For a 1 = a 2 = a 3 the surface is the round 2-sphere about the origin of radius √ a 1 which is totally umbilic. For the rotationally symmetric ellipsoid a 1 = a 2 = a = a 3 , and so the umbilics lie at R = 0 and R → ∞.
To compute the index of the umbilic at R = 0 we find On the other hand, for the triaxial ellipsoid a 1 = a 2 = a 3 , a similar calculation shows that there are 4 umbilic points which lie at θ = 0 and

Denote either of these values by
moves this umbilic to the origin R = 0 and a lengthy computation yields For any section F , define the complex-valued functions G k (θ) by the polar decomposition of∂F :∂ and, in particular, G 0 = A 01 =∂F (0), which vanishes iff ξ = 0 is an umbilic point of S. For such an umbilic point, define the degree N polynomial Here and throughout we assume that A N 1 = 0.
Proposition 2.12. For a non-degenerate isolated umbilic point of order N, the polynomial P N does not have any zeros on the unit circle |ζ| = 1 and the index of the umbilic point is I = K − N/2, where K is the number of zeros of P N that lie inside the unit circle.
Proof. On the unit circle ζ = e 2iθ we have which proves the first statement. For the second statement However, we also have Theorem 2.13. [8] The index of an isolated umbilic point on a convex surface in R 3 is less than or equal to 1.

As a corollary
Corollary 2.14. For a non-degenerate umbilic point of order N on a real analytic surface, the number of zeros of the polynomial P N that lie inside the unit circle is less than or equal to 1 + N/2.
Of course the polynomial P N is not arbitrary: its coefficients satisfy certain relations which are contained in the Lagrangian conditions I to V of Proposition 2.7. To prove our Main Theorem we reverse the preceding argument: given a polynomial P N satisfying the order N Lagrangian conditions, we construct a convex surface in R 3 with isolated umbilic point for which P N is the lowest order term in R of∂F . This we do in the next section.

Proof of the Main Theorem
Our first task is to determine the restrictions placed on the coefficients of the polynomial P N by the Lagrangian conditions I to V. To understand these conditions, define the Fourier decomposition of∂F about the origin bȳ In Figure 1 we decompose each of these Fourier coefficients in powers of R and indicate the Lagrangian conditions placed on the coefficients A nm by enclosing related terms by a box. The four Lagrangian conditions I to IV each has a distinctive shape which enables us to trace this net of relations and assist in calculations. Condition V does not introduce any relationship on the coefficients of∂F as it involves A n0 , which is annihilated by∂. As such, we can view condition V as defining the coefficients A n0 in the full power series after we have determined the other coefficients. Proposition 3.1. For the polynomial P N associated with a non-degenerate umbilic point of order N the Lagrangian conditions I to IV mean that the coefficients must satisfy where N = 2l + 1 or N = 2l + 2.
Proof. Given N , the R N column of Figure 1 lists the lowerest order non-zero coefficients of∂F , namely those that enter into the definition of P N . The Lagrangian condition that these coefficients must satisfy are indicated by the boxes in which they lie. Equations (10) and (11) are obtained by inspection of the boxes that contain only these non-zero terms and lower order terms, which by assumption vanish.
We now show that these relations are precisely the equations (1) given in the Introduction.
Corollary 3.2. For the polynomial P N associated with a non-degenerate umbilic point of order N the Lagrangian conditions I to IV mean that the symmetric functions ∆ N n must satisfy where N = 2l + 1 or N = 2l + 2.
Proof. We have the definitions and so we have the relations For k = N − 2 this reads Thus, with the aid of equation (11) we have Therefore |∆ N N −2 | = 1, as claimed. On the other hand, utilising equation (10) we deduce that Now applying equation (12) again with k replaced by N − k − 2, we get which, after some cancellation, reads Now, given a polynomial P N satisfying relations (1) we seek to construct a convex real analytic surface in R 3 with umbilic point at ξ = 0. We do this by first using the coefficients defined by P N as terms of order N in the power series of∂F , and then adding further terms of higher order in such a way that the full Lagrangian condition holds. By moving to higher order terms we ensure that the index of the umbilic point resides in the lowest order term, namely P N .
Such an extension will be far from unique, but we seek to extend in such a way that minimises the number of non-zero terms introduced. Finally, we must ensure that the resulting umbilic point at ξ = 0 is indeed isolated, and hence we can apply Hamburger's result to prove the stated Theorem.
Once again Figure 1 is useful as a guide to the equations we are dealing with. Suppose then that we have non-zero terms in the R N column of∂F (namely, those given by P N ) and all terms of lower degree are zero. We will see that it is only necessary to go to terms of order R N +2 to complete the Lagrangian conditions. In this we identify 6 situations of overlapping equations that must dealt with separately.  (n + 1)A n+3 m+2 = (m + 1)Ā m+3 n+2 . The first and third of these, rearranged to and substituted in the second, yield and so A m+3 n+2 = n + 1 m + 2Ā n+2 m+1 . With these choices the power series is truncated.

Case 2: III IV IV
First we have the Lagrangian condition at order N : and two order N + 2 conditions: Combining these as before we get

Case 3: II III IV
The Lagrangian condition at order N is and the other conditions are

Case 4: II
Here we have exhausted the Lagrangian condition, but must accommodate the tail of the polynomial P N . That is The other term is Case 5: V As mentioned earlier, this Lagrangian condition does not enter into the relations enjoyed by the coefficients of∂F , but is taken as a definition of the non-vanishing holomorphic part of F . In particular, we have Case 6: I When N = 2l + 2 is even, we have the Lagrangian condition As this involves only one coefficient of the power series it is not strictly necessary to relate it to any other term to maintain the Lagrangian condition. However, in order to ensure that the umbilic on the surface is isolated, we impose the final following condition Combining all of the preceding results Theorem 3.3. Given a polynomial P N satisfying the relations (1), choose A N 1 ∈ C * and define the complex numbers {A n N −n+1 } N −1 n=0 by for n = 1, 2, ...N . Then, for N = 2l + 1 the following function defines a convex real analytic surface S in R 3 : with support function For N = 2l + 2 the function is and the support function is A l+2 l+1 ξ l+2ξl+2 + C.
In both cases, S has an isolated umbilic point at ξ = 0 which is non-degenerate of order N .
Proof. We will show that F and r satisfy (5), which implies the Lagrangian condition (4). For N = 2l + 1 we compute: as claimed.
The proof for N = 2l + 2 is the same with the added computation For N > 0 clearly ξ = 0 is an umbilic point. To show that it is isolated we computē ∂F = (1 + ξξ) By assumption, the polynomial P N has no zeros on the unit circle, and so G N (θ) = 0. Thus R = 0 is an isolated umbilic point and is non-degenerate of order N .

Examples
We now demonstrate direct proofs of the Main Theorem for orders N = 1, 2, 3, 4. For order N ≥ 5 there is no direct method.

N=1,2:
In these cases the statement is easily seen to be trivially true.
The Main Theorem states that such a polynomial has at most 3 roots inside the unit circle. By elementary arguments, as in the previous cases, it is not hard to show that there are at most 4 roots inside the unit circle, but reducing this to 3 roots is extremely difficult by elementary methods.