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The aim of this paper is to introduce the space of roots to study the topological properties of the spaces of polynomials. Instead of identifying a monic complex polynomial with the vector of its coefficients, we identify it with the set of its roots. Viète's map gives a homeomorphism between the space of roots and the space of coefficients and it gives an explicit formula to relate both spaces. Using this viewpoint we establish that the space of monic (Schur or Hurwitz) aperiodic polynomials is contractible. Additionally we obtain a Boundary Theorem.

It is well known that for the stability of a linear system

However, if a continuous or discrete system is modeling a physical phenomenon then it is affected by disturbances. Consequently it is convenient to think that there are uncertainties in the elements of the matrix

To define and study these families, one uses the fact that a real polynomial

The aim of this paper is to present a The aim of this paper is to present a different viewpoint in the study of topological properties of the spaces of Schur and Hurwitz (aperiodic) polynomials which is more natural since the definition of such polynomials is in terms of their roots. Instead of identifying a monic complex polynomial

Using this viewpoint we give simple proofs of some known results about the topology of the spaces of Schur and Hurwitz polynomials; for instance, we give a direct proof (see Section

One could also study a broad variety of families of polynomials directly in the space of roots and via Viète’s map to get the corresponding results in the space of coefficients. In this way we use a topological approach to study the spaces of polynomials. Other works where topological and geometric ideas have been applied in control theory are the papers [

Let

Using this isomorphism we can endow

A (real or complex) polynomial

A (real or complex) polynomial

Following [

Let

Analogously, let

In the same way, let

From the fact that if

In the case of real polynomials we can say more. Since we are mainly interested in real polynomials we shall denote

For the case of real Hurwitz polynomials

Hence we have that

Let

Denote by

Consider the

A point in

Let

Viète’s projective map is a homeomorphism between

Viète’s projective map

Clearly

The restriction

Hence, as a corollary of Theorem

Viète’s map is a homeomorphism between

There are results related to Viète’s Theorems which are consequence of a classical theorem by Maxwell [

Let

What Theorem

We define the following subspaces of

The space of

Clearly its image under Viète’s map is

The space of

Its image under Viète’s map is the set of monic complex Schur polynomials

The space of

Its image under Viète’s map is the set of monic real Schur polynomials

The space of

Its image under Viète’s map is the set of monic Schur aperiodic polynomials

The space of

Its image under Viète’s map is the set of monic complex Hurwitz polynomials

The space of

Its image under Viète’s map is the set of monic real Hurwitz polynomials

The space of

Its image under Viète’s map is the set of monic Hurwitz aperiodic polynomials

As an example of the use of the space of roots

The spaces

It is clear from the definition of the spaces

The boundary

The boundary

By the definition of the spaces

Let

The following theorem is proved in [

The spaces

The following homotopy gives a contraction of

For the case

The following Theorem is indicated in [

The spaces

The following homotopy is a contraction of

For the case when

Recall that a topological space

The spaces

The space

All the homotopy groups of a contractible space are trivial [

The proof for the spaces

Corollary

The spaces

Let

The spaces

Let

In this section we shall study the topology of the spaces of Schur and Hurwitz aperiodic polynomials

The spaces

Since the spaces

The space

The fact that the space

The proofs of the following corollaries are analogous to the proofs of Corollaries

The spaces

The spaces

The spaces

If

If

Theorem

Theorem

Let

From the fact that if

Now we can use the

Define the contraction

Using the space of roots we can give a simpler proof of the homeomorphism between the space of degree

Remember that

We have the following two remarks about previous works.

The proof of Theorem 2.1 in [

A counterexample in dimension

Now with the proof that

The approach of using the roots space

Viète’s map gives an explicit homeomorphism between the space of roots

Also using the space of roots and Viète’s projective map we see that the

The second author would like to thank Carlos Prieto, Ricardo Uribe, and Miguel A. Xicoténcatl for some very helpful conversations. Partially supported by DGAPA-UNAM: PAPIIT IN102208 and CONACYT: J-49048-F. Partially supported by CONACYT by means CB-2010/150532.