Sturm series

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In mathematics, the Sturm series[1] associated with a pair of polynomials is named after Jacques Charles François Sturm.

Definition[edit]

Further information: Sturm chain

Let and two univariate polynomials. Suppose that they do not have a common root and the degree of is greater than the degree of . The Sturm series is constructed by:

This is almost the same algorithm as Euclid's but the remainder has negative sign.

Sturm series associated to a characteristic polynomial[edit]

Let us see now Sturm series associated to a characteristic polynomial in the variable :

where for in are rational functions in with the coordinate set . The series begins with two polynomials obtained by dividing by where represents the imaginary unit equal to and separate real and imaginary parts:

The remaining terms are defined with the above relation. Due to the special structure of these polynomials, they can be written in the form:

In these notations, the quotient is equal to which provides the condition . Moreover, the polynomial replaced in the above relation gives the following recursive formulas for computation of the coefficients .

If for some , the quotient is a higher degree polynomial and the sequence stops at with .

References[edit]

  1. ^ (French) C. F. Sturm. Résolution des équations algébriques. Bulletin de Férussac. 11:419–425. 1829.