# Systems, Structure and Control 2012 Part 11

## Tham khảo tài liệu 'systems, structure and control 2012 part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | On Stability of Multivariate Polynomials 193 A root of p s is a vector s0 s10 sM . sm0 such that p s0 0. If s 2 s . s s20 s30 . s are fixed to some arbitrary value then p S1 s20 . sm0 is an univariate polynomial in the variable s1 of degree n1. In conclusion and in contrast with the univariate case a multivariate p s has a finite number of root manifolds in a n -dimensional complex space. Besides in contrast with the univariate case two multivariate polynomials may be coprime but possessing common roots Kharitonov Torres-Munoz 1999 . Let us denote the set of constant degree m -variate polynomials by Pn p s deg p s n where n n1 n2 . nm n e N is the vector of constant partial degrees. Similar definitions will hold for univariate polynomials. In the analysis of the continuous multivariate polynomials is often used the notion of the conjugate polynomial. The conjugate polynomial of p s with respect to the variable s1 using p s as in the decomposition 2 with respect to the variable s1 is given by p s p - s1 - -sm È a s2 s3 - sm i s1 k 3 k 0 where iikps2. s3 . sm means that all coefficients and variables s2 s3 . sm are changed by their complex conjugates. Clearly the conjugate polynomial can be taken from one until m variables. Hereafter it will be considered unless otherwise stated the conjugate p s with respect to the variable s1 . To distinguish the discrete polynomials from the continuous case and for tradition a discrete multivariate polynomial is notated as q z the variable vector and the coefficient vector used are z z .z .z_ and b fr .fe .b .b. respectively. Besides the z z1 z2 . zmf W00---0 t 10---0 t 20---0 - n -nm p y structure of a discrete polynomial is the same as 1 and it is also possible to write it as in the decomposition 2 . In the analysis of the discrete multivariate polynomials is often used the notion of the reciprocal conjugate polynomial. The reciprocal conjugate polynomial of q z with respect to the variable z1 using q z as in the decomposition 2

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