Zeros of polynomials, solvable dynamical systems and generations of polynomials

Transcription

1 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 1/26 Zeros of polynomials, solvable dynamical systems and generations of polynomials Francesco Calogero Dipartimento di Fisica, Universita di Roma La Sapienza Istituto azionale di Fisica ucleare, Sezione di Roma Abstract Recent findings concerning the relations between the coefficients and the zeros of time-dependent monic polynomials will be reported. They underline a differential algorithm to compute all the zeros of a generic polynomial and allow the identification of novel classes of dynamical systems solvable by algebraic operations, including hierarchies of such ewtonian ( accelerations equal forces ) problems describing an arbitrary number of nonlinearly interacting point-particles moving in the complex plane. And the related notion of generations of (monic) polynomials will be introduced and discussed. Part of this work has been done with Oksana Bihun and with Mario Bruschi.

2 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 2/26 References - [1] F. Calogero, ew solvable variants of the goldfish many-body problem, Studies Appl. Math. 137 (1), (2016); DOI: /sapm [2] O. Bihun and F. Calogero, A new solvable many-body problem of goldfish type, J. onlinear Math. Phys. 23, (2016). - [3] O. Bihun and F. Calogero, ovel solvable many-body problems, J. onlinear Math. Phys. 23, (2016). - [4] O. Bihun and F. Calogero, Generations of monic polynomials such that the coefficients of the polynomials of the next generation coincide with the zeros of polynomial of the current generation, and new solvable many-body problems, Lett. Math. Phys. 106 (7), (2016). - [5] F. Calogero, A solvable -body problem of goldfish type featuring 2 arbitrary coupling constants, J. onlinear Math. Phys. 23, (2016). - [6] F. Calogero, Three new classes of solvable -body problems of goldfish type with many arbitrary coupling constants, Symmetry 8, 53 (2016). - [7] M. Bruschi and F. Calogero, A convenient expression of the time-derivative z n (k) (t), of arbitrary order k of the zero z n (t) of a time-dependent polynomial p (z;t) of arbitrary degree in z, and solvable dynamical systems, J. onlinear Math. Phys. 23, (2016). - [8] F. Calogero, ovel isochronous -body problems featuring arbitrary rational coupling constants, J. Math. Phys. 57, (2016); - [9] F. Calogero, Yet another class of new solvable -body problems of goldfish type, Qualit. Theory Dyn. Syst. (in press). - [10] F. Calogero, ew solvable dynamical systems, J. onlinear Math. Phys. 23, (2016). - [11] F. Calogero, Isochronous -body problem of goldfish type featuring 2 arbitrary rational parameters, onlinearity (submitted to, ). - [12] F. Calogero, Integrable Hamiltonian -body problems in the plane featuring arbitrary functions, J. onlinear Math. Phys. (in press as Letter). - [13] F. Calogero, ew C-integrable and S-integrable systems of nonlinear partial differential equation, Studies Appl. Math. (submitted to, ). - [14] F. Calogero, onlinear differential algorithm to compute all the zeros of a generic polynomial, J. Math. Phys. 57, (4 pages) (2016); arxiv: v1 [math.ca]. - [15] F. Calogero, Comment on onlinear differential algorithm to compute all the zeros of a generic polynomial [J. Math. Phys. 57, (2016)], J. Math. Phys. (4 pages, in press). - [16] O. Bihun and F. Calogero, Generations of solvable discrete-time dynamical systems, J. Math. Phys. (submitted to, ).

3 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 3/26 onlinear differential algorithm to compute all the zeros of a generic polynomial [14] P (z; c ; x ) = z + (c m z m ) = (z x n ) n=1 ow introduce the t-dependent polynomial p (z; γ (t); y (t)) = z + [γ m (t) z m ] = [z y n (t)] n=1 There holds then the following

4 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 4/26 Proposition. Consider the following system of nonlinear first-order differential equations satisfied by the zeros y n (t) of this polynomial: y n(t) = { [y n (t) y s (t)] 1 } s=1,s n {f m(t) [c m γ m (0)] [y n (t)] m }, f m (T) f m (0) = 1, γ m (0) = ( 1) m σ m (y (0)), m = 1,, ; σ m (y ) = 1 s (y y y ) 1 <s 2 < <s m. s1 s2 sm Then: x n = y n (T).

5 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 5/26 It is thus seen that the zeros x n of the polynomial P (z; c ; x ) can be computed---once the coefficients c m of this polynomial have been assigned---via the following procedure. Step one: choose (arbitrarily!) complex numbers y n (0). Step two: compute the quantities γ m (0). Step three: integrate (numerically) the above system of differential equations from t=0 to t=t, starting from the initial data y n (0), getting thereby the values y n (T), which give the sought result, x n = y n (T). [Subcase: f m (t) = t, f m ( t) = 1, T = 1. ]

6 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 6/26 An integrable Hamiltonian -body problem in the plane featuring arbitrary functions [16] { [ξ n (t) ξ j (t)]} ξ n(t) = [ 2 ξ n (t) ξ j (t) ξ n (t) ξ j (t) ] 1 {g m (γ m (t)) [ξ n (t)] m }, γ m (t) = ( 1) m σ m [ξ (t)], σ m (ξ ) = (ξ s1 ξ s2 ξ sm ) 1 s 1 <s 2 < <s m.

7 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 7/26 Coefficients and zeros of monic polynomials p (z; y ; x ) = z + (y m z m ) p (z; y ; x ) = (z x n ) n=1 y is an -vector: its components y m are the coefficients of the polynomial p (z; y ; x ) of degree in the independent (complex) variable z. x is an unordered set of numbers x n, which are the zeros of the polynomial p (z; y ; x ). We denote as x [μ] the -vector, the components x [μ],n of which are given by the specific permutation--- labeled by the integer index μ in the range 1 μ!---of the numbers x n. Hence, to any given x, there generally correspond! different -vectors x [μ]. We will come back to this notion below.

8 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 8/26 Identities satisfied by the coefficients and the zeros of a polynomial y m = ( 1) m σ m (x ), m = 1,, ; σ m (x ) = (x s1 x s2 x sm ) 1 s 1 <s 2 < <s m σ 1 (x ) = x 1 + x 2 + +x ; = 2 σ 1 (x ) = x 1 + x 2 ; σ 2 (x ) = x 1 x 2 = x 2 x 1.

9 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 9/26 An obvious identity (for monic polynomials) [y m (x n ) m ] = (x n ), n = 1,,. Other useful (general) identities σ n,m (x ) = δ n1 + (x s1 x s2 x sn 1 ) ; 1 s 1 <s 2 < <s n 1, s j m, j=1,,n 1 s=1 [( 1) s (x n ) s σ s,m (x )] = δ nm (x n x j ), n, m = 1,,.

10 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 10/26 An important remark σ n,m (x ) = δ n1 + (x s1 x s2 x sn 1 ) 1 s 1 <s 2 < <s n 1 s j m, j=1,,n 1 does not have a clear meaning (because it depends on which is x m ). But the identity [( 1) s (x n ) s σ s,m (x )] = δ nm (x n x j ), s=1 n, m = 1,, does make good sense: this identity holds true for any assignment of the components of the unordered set x.

11 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 11/26 If x = (a, b) then Example: =2 x = (a, b) or x = (b, a) σ 1,1 (x ) = σ 1,2 (x ) = 1, σ 2,1 (x ) = b, σ 2,2 (x ) = a, The identity: n = m = 1, a + b = (a b) n = 1, m = 2, a + a = 0 n = 2, m = 1, b + b = 0 n = m = 2, b + a = (b a). If x = (b, a) : exchange a and b, clearly identity still true.

12 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 12/26 Time-dependent monic polynomials ψ (ζ; γ (t); ξ (t)) = ζ + [γ m (t) ζ m ] ψ (ζ; γ (t); ξ (t)) = [ζ ξ n (t)] n=1 γ (t) is a time-dependent -vector: its components γ m (t) are the coefficients of the time-dependent polynomial ψ (ζ; γ (t); ξ (t)) of degree in the independent (generally complex) variable ζ. ξ (t) is an unordered set of numbers ξ n (t), which are the zeros of the polynomial ψ (ζ; γ (t); ξ (t)) (but they may get ordered via their dependence on time if this dependence is continuous).

13 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 13/26 Two very useful formulas [1] 1 ξ n (t) = { [ξ n (t) ξ j (t)]} {γ m(t) [ξ n (t)] m }, ξ n(t) = [ 2 ξ n (t) ξ j (t) ξ n (t) ξ j (t) ] { [ξ n (t) ξ j (t)]} 1 {γ m ( t) [ξ n (t)] m }, γ m (t) = ( 1) m σ m [ξ (t)]. The second-derivative formula is particularly useful and is referred below as (***).

14 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 14/26 Extension to higher derivatives These formulas have been extended to the third and fourth derivatives in [3], to derivatives of arbitrary order in [7] and to the case of discrete time in [12]. These results are not reported nor discussed in this talk, but they are instrumental to demonstrate the solvable character of some of the dynamical systems reported below.

15 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 15/26 Many new solvable models---including isochronous ones---can be manufactured by taking advantage of these formulas. For instance from (***): { [ξ n (t) ξ j (t)]} ξ n(t) = [ 2 ξ n (t) ξ j (t) ξ n (t) ξ j (t) ] 1 {γ m (t) [ξ n (t)] m }, γ m (t) = ( 1) m σ m [ξ (t)], γ m (t) = ( 1) m n=1 {σ n,m [ξ (t)] ξ n ( t)}. ow assume that a solvable dynamical system of ewtonian type ( acceleration equal force ) reads

16 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 16/26 or equivalently γ (t) = f [γ (t); γ (t); t] γ m(t) = f m [γ (t); γ (t); t], m = 1,,. Then insert these equations of motion in the righthand side of (***). In this manner you obtain a new solvable dynamical system of ewtonian type. Several new examples are exhibited in references [1-13], see above. Here we display only a wellknown example and two of those new examples.

17 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 17/26 The goldfish model γ m(t) = i ω γ m (t), γ m (t) = γ m (0) + i γ m(0) [1 exp(iωt)] ξ n(t) = i ω ξ n (t) + [ 2 ξ n (t) ξ j (t) ξ n (t) ξ j (t) ] Solution: the coordinates ξ n (t) are the roots of ξ n(0) [ ξ ξ n (0) ] = n=1 i ω ω exp(i ω t) 1 Isochronous with period T = 2π/ω---or a, generally small (much less than!), integer multiple of T..,.

18 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 18/26 The goldfish-cm model [2] γ m(t) = ω 2 γ m (t) + 2 g 2 [γ m (t) γ j (t)] 3 { [ξ n (t) ξ j (t)]} j=1,j m ξ n(t) = [ 2 ξ n (t) ξ j (t) ξ n (t) ξ j (t) ] 1 {γ m (t) [ξ n (t)] m }, γ m (t) = ( 1) m σ m [ξ (t)]. Isochronous with period T = 2π/ω---or a, generally small (much less than!), integer multiple of T. ;

19 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 19/26 The goldfish-cm model equations of motion [2] ξ n(t) = [ 2 ξ n (t) ξ j (t) ξ n (t) ξ j (t) ] 1 { [ξ n (t) ξ j (t)]} {{ ω 2 γ m (t) + 2 g 2 [γ m (t) γ j (t)] 3 } [ξ n (t)] m }, j=1,j m γ m (t) = ( 1) m σ m [ξ (t)], σ m (ξ ) = (ξ s1 ξ s2 ξ sm ) 1 s 1 <s 2 < <s m.

20 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 20/26 A solvable -body model featuring 2 arbitrary coupling constants [5] γ m(t) = [A mj γ j (t)] j=1 ξ n(t) = [ 2 ξ n (t) ξ j (t) ξ n (t) ξ j (t) ] ; + { [ξ n (t) ξ j (t)]} { [A mj γ (t)] [ξ j=1 j n (t)] m } ; 1 γ m (t) = ( 1) m σ m [ξ (t)], σ m (ξ ) = (ξ s1 ξ s2 ξ sm ). 1 s 1 <s 2 < <s m

21 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 21/26 These examples are obtained directly from the formula (***). But the process can be iterated over and over again. This motivated the idea to introduce and investigate the generations of monic polynomials obtained by replacing the coefficients of the polynomials of the next generation with the zeros of a polynomial of the previous generation. [4]

22 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 22/26 The seed polynomial p (0) (z; y (0) ; x (0) ) = z + [y m (0) z m ] p (0) (z; y (0) ; x (0) ) = [z x n (0) ] n=1 The first generation of! monic polynomials (μ p 1 ;1) (1) (1) (z; y [μ1 ] ; x [μ1 ] ) = z (1) + [y [μ1 ],m (μ p 1 ;1) (1) (1) (1) (z; y [μ1 ] ; x [μ1 ] ) = [z x[μ1 ],n n=1 y (1) [μ1 ] = x (0) (1) [μ1 ] ; y [μ1 ],m = x [μ1 ],m., z m ], The integer index μ 1 taking values in the range 1 μ 1! labels the permutations of the a priori unordered set x (0). (0). ],

23 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 23/26 The second generation of (!) 2 monic (μ p (2) ;2) (2) (z; y [μ (2) ] polynomials (2) ; x [μ (2) ] (μ p (2) ;2) (2) (z; y [μ (2) ] (2) y [μ (2) ] (1) = x [μ (2) ] (2) ) = z + [y [μ (2) ],m (2) ; x [μ (2) ] n=1 (2) (2) ) = [z x [μ ], (2) ],n (1) = x [μ1 ],[μ 2 ]; y [μ (2) ],m (1) = x [μ (2) ],m z m ], The 2-vector μ (2) = (μ 1, μ 2 ) has integer components μ 1, μ 2 taking values in the range 1 μ 1, μ 2!. The label μ 2 identifies the permutation of the a priori unordered set x [μ1 ] the components of which are the zeros of the polynomial of the previous generation (μ with index μ 1, i. e. the zeros of p 1 ;1) (1) (1) (z; y [μ1 ] ; x [μ1 ] ). (1).

24 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 24/26 The k-th generation of (!) k monic (μ p (k) ;k) (k) (z; y [μ (k) ] (μ p (k) ;k) (k) (z; y [μ (k) ] (k) y [μ (k) ] polynomials (k) ; x [μ (k) ] (k) ) = z + [y [μ (k) ],m (k) ; x [μ (k) ] (k 1) (k 1) = x [μ (k 1) ],[μ k ] = x [μ (k) ] ) = [z x [μ (k) ] ],n z m ], (k) n=1, (k) ; y [μ (k) ],m (k 1) = x [μ (k) ],m The k-vector μ (k) = (μ 1, μ 2,, μ k ) has integer components μ 1, μ 2, μ k taking values in the range 1 μ 1, μ 2,, μ k!. The label μ k identifies the permutation of the a priori unordered set x [ μ (k 1) ] the components of which are the zeros of the polynomial of the (k-1)-th generation with indices μ 1, μ 2,, μ k 1, i. e. the zeros of (μ p (k 1) ;k 1) (k 1) (z; y [μ (k 1) ] (k 1) ; x [μ (k 1) ). ]. (k 1)

25 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 25/26 Question: Why introduce this notion of generations of monic polynomials? Reply: Why not? And note the possibility mentioned above to generate many solvable dynamical systems---including many-body problems of ewtonian type ( accelerations equal forces )---associated with such polynomials. Question to the audience: is this notion of generations of polynomials new? Has it already been investigated?

26 F. Calogero: Solvable dynam. systems and generations of polynomials/ BURGOS,70thRagnisco / / page 26/26 Auguri Orlando!!! Beati i giovani appena settantenni...