A Contraction of the Lucas Polygon

Transcription

1 Western Washington University Western CEDAR Mathematics College of Science and Engineering 4 A Contraction of the Lcas Polygon Branko Ćrgs Western Washington University, brankocrgs@wwed Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Ćrgs, Branko, "A Contraction of the Lcas Polygon" (4) Mathematics 6 This Article is broght to yo for free and open access by the College of Science and Engineering at Western CEDAR It has been accepted for inclsion in Mathematics by an athorized administrator of Western CEDAR For more information, please contact westerncedar@wwed

2 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY S -9939(4)73-4 Article electronically pblished on May, 4 A CONTRACTION OF THE LUCAS POLYGON BRANKO ĆURGUS AND VANIA MASCIONI (Commnicated by N Tomczak-Jaegermann) Abstract The celebrated Gass-Lcas theorem states that all the roots of the derivative of a complex non-constant polynomial p lie in the convex hll of the roots of p, called the Lcas polygon of p We improve the Gass-Lcas theorem by proving that all the nontrivial roots of p lie in a smaller convex polygon which is obtained by a strict contraction of the Lcas polygon of p Based on a simple proof of the classical Gass-Lcas theorem from [5, Theorem 44] and an ineqality from [], we give an improvement of this theorem by showing that all the nontrivial roots of the derivative of a polynomial lie in a smaller convex polygon than predicted by the Gass-Lcas theorem In fact, or reslt is closely related to the following consideration of J L Walsh In [, 34] J L Walsh wrote: A deleted neighborhood of an arbitrary zero of p(z) can be assigned which is known to contain no critical point of p(z) Since no critical point other than a mltiple zero of p(z) can lie on the bondary of the Lcas polygon (assmed non-degenerate), it follows that no critical point lies in a certain strip inside the polygon and bonded by a side of the polygon and by a line parallel to that side We proceed to make this conclsion more precise nder certain conditions: [] However, Walsh s reslts as described in [] improve the Lcas polygon only along sides spanned by two roots of the same mltiplicity and not containing any other root in the interior Also, the formlas and calclations reqired in order to practically compte his improvement seem prohibitive A significant improvement of the Gass-Lcas theorem was achieved by D Dimitrov in [3] The regions that are garanteed to contain all the nontrivial critical points of a polynomial p in [3] are given as intersections of certain nions of circles The main difference between the reslts in [3] and or reslt is that the regions in [3] are not easy to visalize, they are not convex and there is no garantee that these regions are strictly contained in the Gass-Lcas polygon In fact, a calclation of the regions stdied in [3, Corollary ] shows that in none of the cases stdied in Example 3 below are these regions strictly inclded in the corresponding contracted Gass-Lcas polygons constrcted in this note Therefore, the best prediction for the location of the nontrivial critical points of a polynomial is obtained as the intersection of the regions from [3] and or contracted Gass-Lcas polygon Received by the editors October 9, and, in revised form, Febrary, 3 Mathematics Sbject Classification Primary 3C5; Secondary 6C Key words and phrases Roots of polynomials, critical points of polynomials, Gass-Lcas theorem c 4 American Mathematical Society

3 BRANKO ĆURGUS AND VANIA MASCIONI What we set ot to do in this paper (see Theorem ) is to frnish a mch easier procedre that allows s to give a simple specific description of a certain strip from Walsh s qote for any polynomial p with at least two distinct roots, and this strip actally applies to the entire bondary of the Lcas polygon For a historic accont on the Lcas theorem see [6, Section ] For applications of the Lcas theorem in different settings see [, 4, 7, 8, 9,,, 3] We consider complex polynomials By z we denote the complex conjgate of z C and D(, r) denotes a closed disk in C with radis r> and center C Theorem (Gass-Lcas theorem) Let p be a non-constant polynomial with derivative p All the roots of p lie in the convex hll of the roots of p We inclde an explicit proof of this theorem becase this allows s to introdce notation and eqations sefl in the seqel We note that this nsally simple proof, while not common in recent books, is not new: see, for instance, [5, Theorem 44] or [3, Theorem 9] Proof Let ζ be a nontrivial root of p ;thatis,letp (ζ) =andp(ζ) Let w,,w k be all the (distinct) roots of p with mltiplicities m,,m k, respectively It is no restriction to assme that p is of the form k () p(z) = (z w j ) mj j= Taking the derivative of () and dividing by p(z) weget p (z) () p(z) = m j z w j Sbstitting ζ in () and sing the fact that =weobtain = p (ζ) p(ζ) = m j m j = ζ w j ζ w j Ths = j= (3) = wherewept Solving (3) for ζ we get j= j= j= m j ζ w j ζ w j ζ w j = m j ζ w j (ζ w j) j= (4) ζ = j= m j ζ w j (ζ w j)= c j := where (5) d j := c j (ζ w j ), j= m j ζ w j > j =,,k ( ) k j= c c j w j = j j= d j w j, j= c j k µ= c, j =,k µ

4 A CONTRACTION OF THE LUCAS POLYGON 3 Clearly <d j, j=,,k,and k j= d j = Ths,(4)showsthatζ is in the convex hll of the roots of p and the theorem is proved Next we define the qantities that we will need to formlate and prove an improvement of the Gass-Lcas theorem Let p be a polynomial of degree n, n, and assme that p has at least two distinct roots Let Z(p) denotethesetofall roots of p Define ω(p, w) :=min { w v : v Z(p),w v }, w Z(p), ω(p) :=min { w v : w, v Z(p),w v }, Ω(p) :=max { w v : w, v Z(p),w v }, τ(p, w) :=min { w v : v Z(p ),v w }, w Z(p), τ(p) :=min { w v : w Z(p),v Z(p ),v w }, T (p) :=max { w v : w Z(p),v Z(p ),v w } In [, Theorem 4] we proved the following ineqalities: (6) < n ω(p) τ(p) sin(π/n) ω(p) and (7) < m w n ω(p, w) τ(p, w) ω(p, w), sin(π/(n m w )) where m w is the mltiplicity of w Z(p) We will also need the following elementary estimate: (8) T (p) Ω(p) τ(p) To see this, let w Z(p) andζ Z(p ),ζ w, be sch that w ζ = T (p) If l is the line passing throgh ζ and perpendiclar to the segment wζ, then Lcas theorem implies that some v Z(p) exists in the half-plane bonded by l and not containing w Sincev does not lie inside the disk D(v, τ(p)), it follows that Ω w v T (p) + τ(p), and so (8) follows Using (8) and (6) now gives (9) T (p) Ω(p) ω(p) /n Theorem Let p be a non-constant polynomial of degree n with derivative p Let w,,w k be all the (distinct) roots of p with mltiplicities m,,m k, respectively Pt δ = k j= δ j,where and define m j δ j := m j +(n Ω(p) ω(p) ) i j w b := δ δ j w j j=, m iω(p,w i) Then <δ< and all the nontrivial roots of p (that is, the roots of p that are not the roots of p) lie in the Swiss cheese -like region obtained by removing the interiors of all the disks D ( w j,m j ω(p, w j )/n ),j =,,k, from the convex

5 4 BRANKO ĆURGUS AND VANIA MASCIONI polygon which is the contraction of the Lcas polygon of p centered at w and with the contraction coefficient δ Proof Since the contraction centered at w b with the contraction coefficient δ is an affine bijective transformation of the complex plane, it maps the Lcas polygon of p onto the convex hll of the points v j := δw b +( δ)w j, j =,,k To prove that this smaller polygon contains all the roots of p, we give detailed estimates for the coefficients c j and d j,j=,,k, from the proof of the Gass- Lcas theorem Pt ω(p) =ω and Ω(p) = Ω By the definitions and with ζ and d j as in (4) and (5), we have (for every j =,,k) c j () d j = k µ= c = m [ j n µ n m j + ζ w j i j Since by (9), ζ w j T (p) Ω(p) ω(p) /n and by (7), ζ w i τ(p, w i ) m iω(p, w i ), n the eqality () implies m i ζ w i () d j m j +(n Ω(p) ω(p) ) =: δ j i j m iω(p,w i) It follows from () and the definition of d j in (5) (or ()) that () <δ j d j, j =,,k, and k µ= δ µ = δ< Also, for an arbitrary j {,,k} we have (3) d j = d µ δ µ = δ + δ j < µ= µ j m j µ= µ j By (4), () and (3) all the nontrivial roots of p lie in the convex region { } R w := t µ w µ : δ µ t µ δ + δ µ, t µ = µ= µ= Denote by R v the convex hll of the points v j := δw b +( δ)w j = δ µ w µ +( δ + δ j )w j, that is, µ= µ j { R v := s µ v µ : s µ, µ= µ= } s µ = ] j =,,k, To prove that R w = R v we will establish a one-to-one correspondence between the simplices { [t ] T } S w = t t 3 t k t k : δµ t µ δ + δ µ, t µ = µ=

6 A CONTRACTION OF THE LUCAS POLYGON 5 and sch that S v = { [s s s 3 s k s k ] T : sµ, t µ w µ = µ= s µ v µ µ= µ= } s µ = This will be accomplished by solving the system As = t where δ + δ δ δ δ δ δ δ + δ δ δ δ δ 3 δ 3 δ + δ 3 δ 3 δ 3 A = δ k δ k δ k δ + δ k δ k δ k δ k δ k δ k δ + δ k and s = [ ] T, [ ] T s s s 3 s k s k t = t t t 3 t k t k The matrix A is invertible with the inverse δ δ δ δ δ δ δ δ δ δ A = δ 3 δ 3 δ 3 δ 3 δ 3 δ δ k δ k δ k δ k δ k δ k δ k δ k δ k δ k Therefore, the entries of s = A t for t S w are (4) s j = t j δ j δ,j=,,k The eqalities in (4) imply that A maps the simplex S w into the simplex S v Conversely, calclating the components of t = As for s S v we get (5) t j =( δ)s j + δ j,j=,,k, and conclde that A maps the simplex S v into S w ThsAis a bijection between S v and S w This proves that all the nontrivial roots of p lie in the convex hll R v of the points v,,v k, which is the contraction of the Lcas polygon of p centered at w b and with the contraction coefficient δ To see the part of the statement abot Swiss cheese -like region, note that the definition of τ(p, w) implies that none of the disks D(w, τ(p, w)), w Z(p), contains nontrivial roots of p By the ineqality (7) the same is tre for the smaller disks D(w, m w ω(p, w)/n) This completes the proof of the theorem Example 3 Let := + i 3 In this example and in Figres throgh 6 we illstrate Theorem by polynomials with roots of varios mltiplicities at the points,, and Figres throgh 6 show the original Lcas polygon, the shrnken Lcas polygon and the Swiss cheese-like region that contains the roots of the derivative as described in Theorem of the polynomials given in the captions The roots of the polynomials are marked by small black disks and the nontrivial roots of the derivatives are marked by small circles More details for each figre are given below

7 6 BRANKO ĆURGUS AND VANIA MASCIONI z z Figre z 3 Figre z z 3 + z 4 z z z z Figre 3 (z ) 4 (z )(z ) Figre 4 +z + z z 3 z 4 z 5 Figre The polynomial is z 3 with the simple roots,, and as a doble root of the derivative Easy comptations give ω(p) =ω(p, w j )=Ω(p) = 3, and so δ j =/7 and δ =3/7 Since w b =, by Theorem the two nontrivial roots mst lie inside the contraction of the Lcas polygon of p by the factor 4/7 and the Swiss-cheese remark allows s to shave three small circlar triangles from the corners of this smaller region Figre The polynomial is z z 3 + z 4 with the doble root, and single roots and The derivative 3z +4z 3 has roots z = i 8, z = 5 8 i 8, z 3 = We calclate that δ = /75, the contraction coefficient δ = 64/ , and w b =5/37

8 A CONTRACTION OF THE LUCAS POLYGON 7 Figre 3 The polynomial is 3 z +3z z 3 +3z 4 3 z 5 +z 6 with the qadrple root, and single roots and The derivative 3 ( +z) 3 ( +z +z ) has nontrivial roots z = i 4, z = 7 4 i 4 We calclate that δ = 654/663, the contraction coefficient δ = 5969/663 95, and w b =35/9 The distance of the root z to the closest edge of the contracted Lcas polygon is 866 and its distance to the arc next to it is 3437 Figre 4 The polynomial is + z + z z 3 z 4 z 5 with the single root at and doble roots at and Its derivative + z 3 z 4 z 3 5 z 4 has roots z =, z = +, z 3 =, z 4 = We calclate that δ =69/475, the contraction coefficient δ = 46/ , and w b = /3 Figre 5 The polynomial is +i 3 i 3 ( z z 4) + ( 3+i 3 )( z z 5) ( +i 3 ) z 3 +z 6 with a triple root at, a doble root at and a single root at The derivative is (z ) (z ) ( 3 i 3+ ( 3+i 3 ) z +z ) with the nontrivial roots z, = 8 i ( 3 ± i ) i ( i) We calclate that δ = 673/48888, the contraction coefficient δ = , and w b =35 ( i 3 ) / i Figre 6 The polynomial is z z 4 with the single roots at,, and The derivative is 4z 3 with the roots 3, 4 3, We calclate that δ =/7, the contraction coefficient δ =69/7 9783, and w b = Note that the region predicted by Theorem to contain all the roots of the derivative consists of the intersection of the complement of the disk centered at, bonded by the thick circle and the region bonded by the thick line in Figre 6 Example 4 Consider p(z) =z n (z ), and let w =,w = (with mltiplicities m = n,m = ) Easy comptations give ω(p) =ω(p, w j )=Ω(p) = So, in this case we have w b =/, δ = δ = n + and δ = n +

9 8 BRANKO ĆURGUS AND VANIA MASCIONI z z 3 4 Figre 5 (z ) 3 (z ) (z ) Figre 6 z z 4 This translates in a shrnken Lcas polygon that is jst the interval [/(n +), /(n +)] However, the Swiss-cheese part of the theorem in this case gives something sbstantially better than the previos calclations, since shaving off the circles D(, /n) andd(, /n) from[, ] leaves s with the interval [/n, /n], which is of corse better (and optimal, since /n Z(p )) References [] P Borwein, T Erdélyi: Polynomials and polynomial ineqalities, Gradate Texts in Mathematics, 6 Springer-Verlag, 995 MR 97e:4 [] B Ćrgs, V Mascioni: On the location of critical points of polynomials, Proc Amer Math Soc 3 (3), MR 3h:36 [3] D Dimitrov: A refinement of the Gass-Lcas theorem, Proc Amer Math Soc 6 (998), 65-7 MR 98h:35 [4] P Henrici: Applied and comptational complex analysis, Vol, John Wiley & Sons, 988 MR 9d:3 [5] E Hille: Analytic fnction theory, Vol, Ginn and Company, Boston 959 MR :645 [6] M Marden: The Location of the Zeros of the Derivative of a Polynomial, AmericanMathematical Monthly, 4 (935), [7] M Marden: Geometry of polynomials Second edition reprinted with corrections, American Mathematical Society, Providence, RI, 985 MR 37:56 [8] M Mignotte: Mathematics for compter algebra Springer-Verlag, 99 MR 9i:687 [9] M Mignotte, D Ştefănesc: Polynomials An algorithmic approach Springer Series in Discrete Mathematics and Theoretical Compter Science Springer-Verlag, Singapore, 999 MR e: [] G V Milovanović, D S Mitrinović, Th M Rassias: Topics in polynomials: extremal problems, ineqalities, zeros World Scientific Pblishing Co, 994 MR 95m:39 [] A Trowicz: Geometria zer wielomianów (Polish) [Geometry of zeros of polynomials] Państwowe Wydawnictwo Nakowe, Warsaw, 967 MR 37:537

10 A CONTRACTION OF THE LUCAS POLYGON 9 [] J L Walsh: The location of critical points of analytic and harmonic fnctions, Amer Math Soc Colloq Pbl, Vol 34, 95 MR :49d [3] R Webster: Convexity, Oxford University Press, 994 MR 98h:5 Department of Mathematics, Western Washington University, Bellingham, Washington 985 address: crgs@ccwwed Department of Mathematical Sciences, Ball State University, Mncie, Indiana address: vdm@bs-csbsed