Inverse Polynomial Images which Consists of Two Jordan Arcs An Algebraic Solution

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1 Inverse Polynomial Images which Consists of Two Jordan Arcs An Algebraic Solution Klaus Schiefermayr Abstract Inverse polynomial images of [ 1, 1], which consists of two Jordan arcs, are characterised by an explicit polynomial equation for the four endpoints of the arcs. Mathematics Subject Classification 000): 41A10, 30C10 Keywords: Algebraic solution, Inverse polynomial image, Two Jordan arcs, Zolotarev polynomial 1 Introduction Let P n be the set of polynomials with complex coefficients of degree n and let P n P n. Let Pn 1 [ 1, 1]) be the inverse image of [ 1, 1] under the polynomial mapping P n, i.e., P 1 n [ 1, 1]) = { z C : P n z) [ 1, 1] }. 1) In general, Pn 1 [ 1, 1]) consists of n Jordan arcs, on which P n is strictly monotone increasing from 1 to +1, see [13]. If there is a point z 1 C, for which P n z 1 ) { 1, 1} and P nz 1 ) = 0, then two Jordan arcs can be combined into one Jordan arc. This combination of arcs can be seen in a very good way from the inverse image of the classical Chebyshev polynomial T n z) = cosn arccosz)). The inverse image Tn 1 [ 1, 1]) is just [ 1, 1], i.e. one Jordan arc, since there are n 1 points z j with the property T n z j ) { 1, 1} and T nz j ) = 0. Note that T n is, up to a linear transformation, the only polynomial mapping with that property. In this paper, we are interested in polynomials for which the inverse image of [ 1, 1] consists of two Jordan arcs. This property is equivalent to the existence of a certain quadratic equation for the corresponding polynomial. Definition 1. The set {a, b, c, d} of four complex numbers is called a T n -tuple if there exist polynomials T n P n and U n P n such that a quadratic equation sometimes called Pell-equation or Abel-equation) of the form holds, Note that T n and U n are unique up to sign. T n z) Hz) U n z) = 1 ) Hz) := z a)z b)z c)z d). 3) published in: Journal of Approximation Theory ), University of Applied Sciences Upper Austria, Campus Wels, School of Engineering, Austria, laus.schiefermayr@fh-wels.at

2 Inverse polynomial images An algebraic solution In [16, Theorem 3], it was proved that a polynomial T n P n satisfies equation ) if and only if Tn 1 [ 1, 1]) consists of two Jordan arcs with endpoints a, b, c, d. The investigation of quadratic equations of the form ) goes bac to Abel [1] and Chebyshev [5]. Their wor was continued by Zolotarev [1, ] and Achieser [, 3], and in recent times by Peherstorfer [9, 10, 11, 1], Lebedev [6] and Peherstorfer and the present author [16]. For a more detailed history on the subject and some pictures of Tn 1 [ 1, 1]) in several interesting cases, see [16]. Most of the above cited papers mae extensive use of elliptic functions and integrals. Nevertheless, our approach for characterising T n -tuples is purely algebraic and is based on the following result of Peherstorfer and the author [14, Lemma.1]. Lemma 1 Peherstorfer and Schiefermayr [14]). i) Let n = m + 1 be an odd degree. The set {a, b, c, d} is a T n -tuple if and only if it satisfies the following system of equations: x x ) m y ym 1 ) + 1 a b c + d ) = 0, = 1,,..., m. 4) The corresponding polynomial T n is given by T n z) = 1 z d) m z x j) a d) m a x j) = 1 + z a)z b)z c) m 1 z y j) d a)d b)d c) m 1 d y j). Note that T n x j ) = T n d) = 1 and T n y j ) = T n a) = T n b) = T n c) = 1. ii) Let n = m + be an even degree. The set {a, b, c, d} is a T n -tuple if and only if it satisfies one of the following two systems of equations: x x ) m y ym ) + 1 a b + c + d ) = 0, = 1,..., m ) or x x ) m+1 y ym 1 ) + 1 a b c d ) = 0, = 1,..., m + 1. The corresponding polynomial T n for the solution of 5) is given by 6) T n z) = 1 z c)z d) m z x j) a c)a d) z a)z b) m m a x = 1 + z y j) j) c a)c b) m c y j). If {a, b, c, d} satisfies system 6), i.e., is a T n -tuple, then {a, b, c, d} is also a T n -tuple and for the corresponding polynomial we have T n z) = T n z) 1. Note that the x j and y j of equation 4), 5), and 6), are the zeros of the corresponding polynomial U n and are therefore exactly the extremal points of the corresponding polynomial T n on Tn 1 [ 1, 1]) which consists of two Jordan arcs). As usual, a point z 0 C, C C compact, is called an extremal point of P n P n on C if P n z 0 ) = max z C P n z).

3 Inverse polynomial images An algebraic solution 3 Further, we want to remar that the above definition of a T n -tuple also includes the case of one interval if two of the four points a, b, c, d are equal). The main purpose of the present paper is to modify the polynomial systems 4) and 5) in the following way: With the help of the recent paper [0], in Theorem 1 and Theorem, we give one polynomial equation in terms of a, b, c, d, which is equivalent to 4) and 5), respectively. In other words, for every degree n, we can explicitly give a polynomial in four variables pa, b, c, d), whose zeros {a, b, c, d} are T n -tuples. Moreover, a simple equation for computing the extremal points x j and y j is derived. Algebraic solutions of the quadratic equation ) with the help of Jacobi s elliptic functions can be found in [17, 18] in the real case), and [16, Section 4]. In [9, Section 5], an algebraic solution in the real case) is given with the help of orthogonal polynomials. In [15, Chapter 7], an algebraic solution was found by simplifying the above system of equations with the help of Gröbner-Basis. The paper is organised as follows. In section, the fundamental lemma based on [0] is given, from which the simplifications of 4) and 5), proved in section 3, can be deduced. Moreover, the maximum number of T n -tuples is given explicitly assuming that 3 of the 4 points a, b, c, d are fixed. In section 4.1, the polynomial equations for a, b, c, d and for x j and y j are explicitly written down for the smallest degrees n {, 3, 4}. Finally, a brief loo at the special case of Zolotarev polynomials is taen in Section 4.. Auxiliaries Let us define F 0 := 1 and, for = 1,,..., F F s 1, s,..., s ) := 1)! det s s s s 3 s s s 1 s s 3 s 4... s 1 1 s s 1 s s 3... s s 1. 7) For negative indices = 1,, 3,..., we define F := 0. The next lemma, which is fundamental for our considerations, can be extracted from [0]. Lemma. For given s 1, s,..., s ν+µ C, consider the system of equations u u ) ν v vµ ) = s, = 1,,..., ν + µ. 8) If {u 1,..., u ν, v 1,..., v µ } is a nontrivial solution of 8), i.e., the sets {u 1,..., u ν } and {v 1,..., v µ } are disjoint, then this solution is unique up to permutations of the u j and v j ) and the values v 1, v,..., v µ are exactly the solution of the equation v µ + Λ 1 v µ 1 + Λ v µ Λ µ 1 v + Λ µ = 0, 9)

4 Inverse polynomial images An algebraic solution 4 Λ 1, Λ,..., Λ µ are the solution of the following regular linear system of equations: µ F ν+1 i Λ i = F ν+1 i=1 µ F ν+ i Λ i = F ν+ i=1 10). µ F ν+µ i Λ i = F ν+µ i=1 By Cramer s rule, the solution Λ 1, Λ,..., Λ µ of system 10) may be written in the form note that system 10) is regular and therefore det F 0) and Λ i = det F i, i = 1,,..., µ, 11) det F F ν F ν 1 F ν... F ν+1 µ F ν+1 F ν F ν 1... F ν+ µ F := F ν+ F ν+1 F ν... F ν+3 µ R µ µ 1) F ν+µ 1 F ν+µ F ν+µ 3... F ν F i := F and the i-th column of F is replaced by By 11), equation 9) may be written in the form F ν+1 F ν+ F ν+3. F ν+µ. 13) v µ det F + v µ 1 det F 1 + v µ det F v det F µ 1 + det F µ = 0. 14) 3 Main Results Let n = m + 1 be an odd degree. Starting point is system of equations 4), which may be written in the form y ym 1 ) x x m + d ) = s, = 1,,..., m, 15) s := 1 a b c d ), = 1,,..., m. 16) Then, by Lemma, the above polynomial system reduces to d m+1 det F + d m det F 1 + d m 1 det F d det F m + det F m+1 = 0, s, F, F, and F i is defined in 16), 7), 1), and 13), respectively, and ν = m 1 and µ = m + 1. Note that if {y 1,..., y m 1, x 1,..., x m, d} is a solution of 15) then the

5 Inverse polynomial images An algebraic solution 5 sets {y 1,..., y m 1 } and {x 1,..., x m, d} are disjoint, thus, by Lemma, the corresponding linear system 10) is regular and det F 0. In order to get a polynomial equation for the y i, we write system 4) in the form x x m + d ) y ym 1 ) = s, = 1,,..., m, 17) s := 1 a + b + c + d ), = 1,,..., m. 18) Then, again by Lemma, the above polynomial system reduces to y m 1 det F + y m det F 1 + y m 3 det F y det F m + det F m 1 = 0, s, F, F, and F i is defined in 18), 7), 1), and 13), respectively, and ν = m + 1, µ = m 1. By the same argument as above, det F 0. For a polynomial equation for the x i, we write system 4) in the form y ym 1 + a ) x x ) m = s, = 1,,..., m, 19) s := 1 a b c + d ), = 1,,..., m. 0) Then, again by Lemma, the above polynomial system reduces to x m det F + x m 1 det F 1 + x m det F x det F m 1 + det F m = 0, s, F, F, and F i is defined in 0), 7), 1), and 13), respectively, and ν = m, µ = m. By the same argument as above, det F 0. We collect the above results in the following theorem. Theorem 1. Let n = m + 1. i) The set {a, b, c, d} is a T n -tuple if and only if a, b, c, d satisfies the polynomial equation p := d m+1 det F + d m det F 1 + d m 1 det F d det F m + det F m+1 = 0, 1) s, F, F, and F i is defined in 16), 7), 1), and 13), respectively, and ν = m 1, µ = m + 1. ii) The values y 1, y,..., y m 1 are exactly the zeros of the polynomial y m 1 det F + y m det F 1 + y m 3 det F y det F m + det F m 1, ) s, F, F, and F i is defined in 18), 7), 1), and 13), respectively, and ν = m + 1, µ = m 1. iii) The values x 1, x,..., x m are exactly the zeros of the polynomial x m det F + x m 1 det F 1 + x m det F x det F m 1 + det F m, 3) s, F, F, and F i is defined in 0), 7), 1), and 13), respectively, and ν = m, µ = m. Corollary 1. Let n = m + 1.

6 Inverse polynomial images An algebraic solution 6 i) The polynomial p pa, b, c, d) in Theorem 1i) is a homogeneous polynomial of a, b, c, d with rational coefficients and degree m + m = n 1)/4. ii) Let 3 of the 4 points a, b, c, d C be fixed, then there exist at most n 1) T n -tuples containing these 3 points. Proof. i) By the definitions 16), 7), 1), and 13), the following statements concerning the degree of p hold note that ν = m 1, µ = m + 1): F is a homogeneous polynomial of a, b, c, d with degree. det F is a homogeneous polynomial of a, b, c, d with degree m 1. det F i is a homogeneous polynomial of a, b, c, d with degree m 1 + i. From these statements, the assertion follows. ii) By the special form of equation 4), there are 4 different possibilities to fix 3 of the 4 points a, b, c, d in 4). These 4 possibilities multiplied with the degree n 1)/4 of the homogeneous polynomial pa, b, c, d) gives the maximum number of different solutions. Finally, we give the analogous results for even degree. Since the proofs run along the same lines as those for odd degree, we omit them. Theorem. Let n = m +. i) The set {a, b, c, d} is a T n -tuple but not a T n -tuple if and only if a, b, c, d satisfies the polynomial equation p := a m+1 det F + a m det F 1 + a m 1 det F a det F m + det F m+1 = 0, 4) s := 1 a + b c d ), = 1,,..., m + 1, 5) and F, F, and F i is defined in 7), 1), and 13), respectively, and ν = m, µ = m + 1. ii) The values y 1, y,..., y m are exactly the zeros of the polynomial y m det F + y m 1 det F 1 + y m det F y det F m 1 + det F m, 6) s := 1 a + b + c d ), = 1,,..., m + 1, 7) and F, F, and F i is defined in 7), 1), and 13), respectively, and ν = m + 1, µ = m. iii) The values x 1, x,..., x m are exactly the zeros of the polynomial x m det F + x m 1 det F 1 + x m det F x det F m 1 + det F m, 8) s := 1 a b + c + d ), = 1,,..., m + 1, 9) and F, F, and F i is defined in 7), 1), and 13), respectively, and ν = m + 1, µ = m.

7 Inverse polynomial images An algebraic solution 7 Corollary. Let n = m +. i) The polynomial p pa, b, c, d) in Theorem i) is a homogeneous polynomial of a, b, c, d with rational coefficients and degree m + 1) = n /4. ii) Let 3 of the 4 points a, b, c, d C be fixed, then there exist at most 3n /4 T n -tuples containing these 3 points. 4 Addenda 4.1 Equations for Small Degrees By Theorems 1 and, the computation of a T n -tuple {a, b, c, d} and the extremal points x i and y i of the corresponding polynomial T n can be managed in 4 steps: 1. Given 3 of the 4 points a, b, c, d, compute the 4 th point by equation 1) and 4), respectively.. Compute the y j by equation ) and 6), respectively. 3. Compute the x j by equation 3) and 8), respectively. 4. Compute the corresponding polynomial T n by Lemma 1. In the following, we give the equations for a, b, c, d, for the y j, and for the x j, in case of the simplest degrees n =, 3, 4. For greater degrees, the equations get very buly. n = : a + b c d = 0 n = 3: a ab + b ac bc + c + ad + bd + cd 3d = 0 x 4a + 4b + 4c 4d) + a + ab 3b + ac bc 3c ad + bd + cd + d ) = 0 n = 4: a 4 + 4a 3 b 10a b + 4ab 3 + b 4 4a 3 c + 4a bc + 4ab c 4b 3 c + 6a c 4abc + 6b c 4ac 3 4bc 3 + c 4 4a 3 d + 4a bd + 4ab d 4b 3 d 4a cd 8abcd 4b cd + 4ac d + 4bc d + 4c 3 d + 6a d 4abd + 6b d + 4acd + 4bcd 10c d 4ad 3 4bd 3 + 4cd 3 + d 4 = 0 y a + 4ab b + 4ac + 4bc c 4ad 4bd 4cd + 6d ) + a 3 a b ab + b 3 a c + abc b c ac bc + c 3 + a d abd + b d acd bcd + c d + 3ad + 3bd + 3cd 5d 3 ) = 0 x a 4ab + 6b + 4ac 4bc c + 4ad 4bd + 4cd d ) + a 3 + a b + 3ab 5b 3 a c abc + 3b c ac + bc + c 3 a d abd + 3b d + acd bcd c d ad + bd cd + d 3 ) = 0

8 Inverse polynomial images An algebraic solution 8 4. Special Case: Zolotarev Polynomial A special case for which the inverse polynomial image consists of two arcs is the socalled Zolotarev polynomial, which has also applications in signal processing [19]. Given σ > 0, the Zolotarev polynomial Z n x) solves the following approximation problem [4, Addendum E]: min max x n nσx n 1 + a n x n a 1 x + a 0 = max Zn x) =: Ln a j C x [ 1,1] x [ 1,1] It is well nown that, for σ tan π n, the Zolotarev polynomial Z nx) is simply a suitable linear transformed classical Chebyshev polynomial T n x) and, for σ > tan π n ), the inverse image of Z n x) consists of the two arcs [ 1, 1] [α, β], 1 < α < β and Z n has n and extremal points in [ 1, 1] and [α, β], respectively. The connection of the parameter σ with the extremal points of Z n follows from the well nown theorem of Vieta applied to the polynomial Z n x) + L n, which in our notation put a = α, b = 1, c = 1, d = β) reads as follows: y j + α = nσ n = m + 1) 30) m 1 m y j + α + 1 = nσ n = m + ) 31) We summarize the results in the following corollary. Corollary 3. Given n N and σ > tan π n ), the two endpoints α and β of the inverse polynomial image of the corresponding Zolotarev polynomial can be computed by the following two polynomial equations: i) n = m + 1: β m+1 det F + β m det F 1 + β m 1 det F β det F m + det F m+1 = 0 3) F, F, F i is defined in 7), 1), 13), respectively, and ii) n = m + : det F 1 + α nσ) det F = 0 33) ν = m 1, µ = m + 1, s := 1 α 1 1) β ) in 3) ν = m + 1, µ = m 1, s := 1 α ) + β ) in 33) α m+1 det F + α m det F 1 + α m 1 det F α det F m + det F m+1 = 0 34) F, F, F i is defined in 7), 1), 13), respectively, and det F 1 + α + 1 nσ) det F = 0 35) ν = m, µ = m + 1, s := 1 α + 1 1) β ) in 34) ν = m + 1, µ = m, s := 1 α ) β ) in 35) Proof. Equation 3) and 34) follow immediately by Theorem 1i) and Theorem i), respectively. Equation 33) and 35) follow from 30) and 31) together with Theorem 1ii) and Theorem ii), respectively, using Vieta s root theorem.

9 Inverse polynomial images An algebraic solution 9 Remar 1. i) Let n N and σ > tan π n ). With the equations 3),33), and 34),35), respectively, one can determine α, β uniquely such that 1 < α < β. ii) Once α, β are determined, by Theorem 1ii),iii) and Theorem ii),iii), respectively, one can easily compute the extremal points x j and y j which are all in [ 1, 1]). iii) With α, β, the x j, and the y j, the corresponding Zolotarev polynomial ist given, for n = m + 1 odd, Z n x) = x β) m m x x j ) + 1 β α) α x j ) m 1 = x α)x 1) x y j ) 1 β α)β 1) β y j ), m 1 36) and, for n = m + even 1, m m Z n x) = x + 1)x β) x x j ) + 1 α + 1)β α) α x j ) = x 1)x α) m m x y j ) 1 β 1)β α) β y j ). 37) iv) For a different approach to an algebraic solution of the Zolotarev problem, see [7, 8]. References [1] N.H. Abel, Sur l intégration la formule différentielle..., Journal Reine Angew. Math ), in French). [] N.I. Achieser, Über einige Funtionen, die in gegebenen Intervallen am wenigsten von Null abweichen, Bull. Phys. Math., Kasan, Ser.III 3 199), 1 69 in German). [3], Über einige Funtionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen, Bull. Acad. Sci. URSS 7 193), in German). [4], Theory of approximation, Dover Publications, 199. x+a [5] P.L. Chebyshev, Sur l integration de la differentielle dx, Oevres x 4 +αx 3 +βx +γx+δ de P.L. Tchebychef, chapitre 1, Chelsea Publ. Co., New Yor 1961). [6] V.I. Lebedev, Zolotarev polynomials and extremum problems, Russian J. Numer. Anal. Math. Modelling ), [7] V.A. Malyshev, The Abel equation, St. Petersburg Math. J ), In the original article, there is an error in the second part of formula 37), which is corrected here. I am very grateful to Prof. Heinz-Joachim Rac for pointing out this mistae.

10 Inverse polynomial images An algebraic solution 10 [8], Algebraic solution of the Zolotarev problem, St. Petersburg Math. J ), [9] F. Peherstorfer, Orthogonal and Chebyshev polynomials on two intervals, Acta Math. Hungar ), [10], On Bernstein-Szegö orthogonal polynomials on several intervals. II. Orthogonal polynomials with periodic recurrence coefficients, J. Approx. Theory ), [11], Orthogonal and extremal polynomials on several intervals, J. Comput. Appl. Math ), [1], Elliptic orthogonal and extremal polynomials, Proc. London Math. Soc. 3) ), [13], Inverse images of polynomial mappings and polynomials orthogonal on them, J. Comput. Appl. Math ), [14] F. Peherstorfer and K. Schiefermayr, Description of extremal polynomials on several intervals and their computation I, Acta Math. Hungar ), [15], Description of extremal polynomials on several intervals and their computation II, Acta Math. Hungar ), [16], Description of inverse polynomial images which consist of two Jordan arcs with the help of Jacobi s elliptic functions, Comput. Methods Funct. Theory 4 004), [17] M.L. Sodin and P.M. Yuditsiĭ, Functions that deviate least from zero on closed subsets of the real axis, St. Petersburg Math. J ), [18], Algebraic solution of a problem of E.I. Zolotarev and N.I. Ahiezer on polynomials with smallest deviation from zero, J. Math. Sci ), [19] M. Vlče and R. Unbehauen, Zolotarev polynomials and optimal FIR filters, IEEE Trans. Signal Process ), [0] Y. Wu and C.N. Hadjicostis, On solving composite power polynomial equations, Math. Comp ), electronic). [1] E.I. Zolotarev, Sur la méthode d integration de M. Tchebichef, Journal Math. Pures Appl ), [], Applications of elliptic functions to problems of functions deviating least and most from zero in Russian), Oeuvres de E.I. Zolotarev, Vol., Izdat. Aad. Nau SSSR, Leningrad 193), 1 59.

Inverse Polynomial Images and Inequalities for Polynomials Minimal with respect to the Maximum Norm HABILITATION THESIS

Inverse Polynomial Images and Inequalities for Polynomials Minimal with respect to the Maximum Norm HABILITATION THESIS submitted to the Faculty of Engineering and Natural Sciences Johannes Kepler University