SOME RESULTS ON THE GEOMETRY OF THE ZEROS OF POLYNOMIALS

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Serie A, OF THE ROMANIAN ACADEMY Volume 2, Number /20, pp. 6 2 SOME RESULTS ON THE GEOMETRY OF THE ZEROS OF POLYNOMIALS J. L. DÌAZ-BARRERO, J. J. EGOZCUE ad P. G. POPESCU Applied Mathematic III Polytechical Uiverity of Cataloia Jordi Giroa -3, C2, Barcelo, Spai Applied Sciece Faculty, Politehica Uiverity, Bucureşti, Româia pgpopecu@yahoo.com A eceary ad ufficiet coditio for a polyomial with complex coefficiet to have all it ero o the uit circle, geeraliig a claical reult etablihed by Schur i 97, i preeted. Furthermore, icomplete polyomial are reviited ad a a coequece a geeraliatio of a theorem of Coh o locatio of ero of polyomial i alo give Key word: Geometry of polyomial; Locatio of ero; Covex hull; Reflectio coefficiet; Uit circle problem.. INTRODUCTION The tudy of polyomial from a o-algebraic tadpoit i to a great extet coected with the geometric theory of fuctio of a complex variable. Thi coectio become clear whe we examie the type of problem dealt with ad the techique ued to olve them. Hereof, ome reult i the Geometry of the Zero of a Polyomial i a Complex Variable" recetly publihed are reviited. They are ued to etablih ew oe ad to geeralie claic reult o locatio of the ero of polyomial. A baic cocept cotiuouly tae ito accout whe derivig thee reult i the characteriatio of polyomial by their reflectio coefficiet, obtaied by uig Schur-Coh type recurio ([, 2]). It ca be foud i [7] ad the referece therei, where ome reult o locatio of the ero are preeted. Polyomial characteried by reflectio coefficiet appear frequetly i may area of applied ciece ad techology ([, 0, 5]). Example ca be foud i electric egieerig, eimology ad i cotrol theory applicatio ([9, 3, 3]). A firt goal i thi paper i to obtai coditio for a polyomial to have all it ero o the uit circle. A a coequece, a geeraliatio of a claical reult preeted by Schur i 97 i alo give. A ecod goal i to reviit icomplete polyomial, itroduced ad efficietly ued to obtai a geeraliatio of the well-ow Gau-Luca theorem i [8], i order to geeralie aother claical reult etablihed by Coh i NOTATION AND BASIC CONCEPTS Some otatio ad cocept that will be ued hereafter are recalled. Let A be a moic polyomial with complex coefficiet of degree, amely, A a. = + The reciprocal polyomial A ( ) of A i defied by = 0 = =, = 0 = 0 = A A (/ ) a a ( ),

2 2 Reult o the geometry of the ero 7 where a deote the complex cougate of a. It ero, = /, are the ivere of the ero of A ( ) i the uit circle. A immediate coequece of the well-ow fact that A = A for all with = (ee [2]), i that if A ( ) ha o ero of modulu ; the A ( ) ha o ero o the uit circle =. I thi cae, if A ( ) ha m,( m ) ero iide the uit circle, the A ( ) ha m ero i < The reflectio coefficiet α ', alo ow i the literature a Schur-Segö parameter [5] or partial correlatio (PARCOR) coefficiet [0] ca be obtaied from A by uig bacward Levio' recurio ([, 5]). A A A ( ) = ( ) ( ), 2 α a where α = a0. From () ad ome ad ome traightforward algebra forward Levio' recurio A = A +α A, (2) i obtaied. Now, for eae of referece, we give the defiitio of the characteriatio of A by it reflectio coefficiet a give i [7]. Defiitio. The characteriatio of a moic complex polyomial A uig reflectio coefficiet i give by A = A( ); α +, α + 2,..., α, where A, called bae polyomial, i either A = A0 = for = 0, or A i a o-elfiverive uitary polyomial (i.e., a 0 = ) for. The α C, = +, + 2,,, are reflectio coefficiet. () 2. A GENERALIZATION OF A THEOREM OF SCHUR Before dealig with the mai reult of thi ectio, ome baic reult will be tated. We begi with a Lemma o Blache' partial product. Lemma. Let ξ, w ( ξ ) =. ξ,, ad ξ be complex umber. If ad ξ <, the w ( ξ ) <,where, < ξ, Proof. We have to prove that w ( ξ ) = <, which i equivalet to provig that ξ, ξ < ξ, or,, 2 2 ξ < ξ, for, < ad ξ <. Let u deote by:,, ad by 2 2 2,,,, A=ξ =ξ ξ ξ ,,,, B= ξ = ξ ξ + ξ,

3 8 J.L. Dìa-Barrero, J.J. Egocue, P.G. Popecu 3 repectively. The 2 2 B A= (, )( ξ ) > 0 ad the lemma i proved. A ey ad well-ow reult i the theory of polyomial ad reflectio coefficiet ([, 2, 4, 7]), that we will ue further o, i Theorem. Let A = = a be a moic complex polyomial with reflectio coefficiet 0 α, α2,, α. The A ha all it ero iide the uit circle if ad oly if α < for =,2,,. Aother importat reult, publihed i [6], that will be eeded, i Theorem 2. The polyomial whoe characteriatio with the reflectio coefficiet i of the form [ α, α2,, α ] with α <,, ad α = have all their ero o the uit circle ad they are imple. Now, we tate ad prove our mai reult. It give a eceary ad ufficiet coditio for a polyomial to have all it ero o the uit circle. Theorem 3. Let A be a moic polyomial with complex coefficiet. It ha all it ero o the uit circle if ad oly if there exit a polyomial A α, α2,..., α, α <, =,2,...,, uch that A α, α2,..., α, 0,..., 0, α, α = for ome 0. Proof. ) If A ha all it eroe o =, the A i elf iverive. That i, A =α A, α =. The, taig = 0 ad A = A, we have: 2 A +α A = A + α A = A + A = A ( ) ( ) ( ) ( ) ( ) ( ) ( ) ad the eceary coditio i proved. ) By applyig Theorem ad Theorem 2 to the polyomial A, we have that all it ero lie iide or o the uit circle. Hece, A = = ( ) ad A = ( ). = Now we claim that A oly ha uitary eroe. I fact, let ξ be a ero of. The A ξ =ξ A ξ +α A ξ = ad ξ A ( ξ ) = α A ( ξ ). That i, ( ) ( ) ( ) 0 A. (3) = ξ ξ = ξ Sice = whe =, the i (3) thee factor implify ad we oly have to coider factor with <. Aume that ξ>. The, by Lemma, we have

4 4 Reult o the geometry of the ero 9 ad ξ > ξ, = = = = = ξ a ξ > ξ ξ > ξ, which cotradict (3). O the other had, if ξ <, agai o accout of Lemma, we have ad ξ < ξ,, = = = = = ξ ξ < ξ ξ < ξ, i cotradictio with (3). Therefore, (3) hold oly whe ξ =. Thi lead u to the cocluio that all the ero of A lie o the uit circle. Thu the ufficiet coditio i etablihed ad the theorem i proved. Fially, we poit out that the precedig theorem i a geeraliatio of a reult etablihed by Schur i 97 [6]. It wa tated a Corollary. Let A be a moic polyomial with complex coefficiet whoe ero lie withi the uit circle. The A ( ) A + ha all it ero o the uit circle. More geerally, the root of the equatio A +λ A = 0 lie outide the uit circle for λ <, iide for λ > ad o it boudary for λ =. 3. A GENERALIZATION OF A THEOREM OF COHN I [8] icomplete polyomial were defied ad ued to geeralie the well-ow Gau-Luca theorem. I thi ectio icomplete polyomial will be coidered agai, to geeralie a reult of Coh o locatio of the ero of polyomial ad it derivative. We begi with Defiitio 2. Let A be a moic polyomial with complex coefficiet ad ero, 2,,. A polyomial A ( ) i called a covex liear combiatio of icomplete polyomial aociated with A if A ( ), alo deoted by A, i give by where ( ) ad 2 = = A g, =,,..., i a vector which compoet are o egative real umber uch that g = ( ) are the icomplete polyomial. = =, Now, we tate a reult preeted i [8] dealig with the tudy of the covex hull of the ero of icomplete polyomial. It will be ued later o. =

5 20 J.L. Dìa-Barrero, J.J. Egocue, P.G. Popecu 5 Theorem 4 (Gau-Luca geeralied). Let, 2,, be ot ecearily ditict, complex umber. The, the polyomial = = H,,, ) of the ero of A = ( ). ( 2 The mai reult i thi ectio i A g ha all it ero i or o the covex hull = Theorem 5. The polyomial A ha all it ero o = if ad oly if there exit a covex liear combiatio of icomplete polyomial, aociated with A, ay ( ), uch that it geerate A A by forward recurio (2) with a uitary reflectio coefficiet ad ha all it ero iide or o the uit circle. Proof. Firt, we prove the eceary coditio. I fact, if A ( ) ha all it ero i ad we apply forward recurio (2) with α uitary, that i, α = a0. The, o accout of Theorem 2, A ha all it ero o =. I order to prove the ufficiet coditio, we aume that A ha all it ero o = : the, by Theorem 4, every covex liear combiatio of icomplete polyomial, amely, = = A = A = g, =, 0, ha all it ero i (,,..., 2 ) U = { C : } H ad the theorem i proved. Corollary (Coh' Theorem). A ha all it ero o the uit circle if ad oly if A i elfiverive ad A' ha all it ero iide or o the uit circle. we have: Proof. Sice A i elf-iverive, a 0 = α =. By Theorem 5, if we chooe ad the reult i proved. ' ( ) = ( ) = ( ), =, = = A g A =,, ACKNOWLEDGEMENTS The firt two author tha Miitry of Sciece ad Iovatio of Spai that ha upported thi reearch by grat MTM , ad the third author would lie to tha the Romaia Miitry of Labour, Family ad Social Protectio, that ha upported thi reearch through POSDRU/6/.5/S/9. REFERENCES. K. J. ASTROM, Itroductio to Stochatic Cotrol Theory, Academic Pre, J. P. BURG, Maximum Etropy Spectral Aalyi, PhD Thei, Staford Uiverity, Staford, CA., J. F. CLAERBOUT, Fudametal of Geophyical Data Proceig with Applicatio to Petroleum Propectig, Bael, Birhauer-Verlag, A. COHN, Über die Aahl der Wurel eier algebraiche Gleichug i eiem Kreie, Math. Z, 4, 0 48 (992).

6 6 Reult o the geometry of the ero 2 5. T. CONSTANTINESCU, Schur Parameter, Factoriatio ad Dilatio Problem, McGraw-Hill, New Yor, J. L. DÌAZ-BARRERO, O the ditributio of the ero of a complex polyomial, Aalele Uiverităţii Bucureşti, LVIII,, (2009). 7. J. L. DÌAZ-BARRERO ad J. J. EGOZCUE, Characteriatio of Polyomial Uig Reflectio Coefficiet, Applied Mathematic E-Note, 4, 4 2 (2004). 8. J. L. DÌAZ-BARRERO ad J. J. EGOZCUE, A geeraliatio of Gau-Luca Theorem, Cecholova Mathematical Joural, 58, (2008). 9. T. KAILATH, A View of Three Deacade of Liear Filterig Theory, IEEE Tra. o Iformatio Theory, IT-20, S. M. KAY, Moder Spectral Etimatio, Pretice Hall, New Jerey, N. LEVINSON, The Wieer rm (root mea quare) error criterio i filter deig ad predictio, J. Math. Phy., 25, (947). 2. M. MARDEN, Geometry of Polyomial, Mathematical Survey Number 3, America Mathematical Society, Providece Rhode Ilad, U. NURGES, Robut pole aigmet via reflectio coefficiet of a polyomial, Automatica, 42, (2006). 4. A. PAPOULIS, Probability, Radom Variable ad Stochatic Procee, McGraw Hill, Sigapore, B. PICINBONO ad M. BENIDIR, Some Propertie of Lattice Autoregreive Filter, IEEE Tra. o Acout. Speech Sigal Proce, ASSP 34, (986). 6. I. SCHUR, Über Potereihe, die im Ier de Eiheitreie bechrät id, J. Reie Agew. Math., 47, (97). 7. S. TREITEL, T. J. ULRYCH, A ew Proof of the Miimum-Phae Property of the Uit Predictio Error Operator, IEEE Tra. o Acoutic, Speech ad Sigal Proceig, ASSP-27,, (979). Received December 23, 200