       Approximate/asymptotic factorization

• To: mathgroup at smc.vnet.net
• Subject: [mg73509] Approximate/asymptotic factorization
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Wed, 21 Feb 2007 01:35:21 -0500 (EST)
• Organization: The University of Western Australia

```Consider the monomial H[4,4]

256*(23625 + 126000*z^4 - 7200*z^8 + 3840*z^12 + 256*z^16)

where H[m,n] is a Generalized Hermite Polynomial (Noumi & Yamada 1998),
a rational solution to the Painlevé equation P_IV. A plot of the roots
of H[4,4] displays an interesting quasi-rectangular structure.

Then consider the monomial Q[4,4]

1528899609315374375625 + 6795109374734997225000*z^4 -
560866170613047390000*z^8 + 153399294526645440000*z^12 +
2734590598399296000*z^16 - 167891551278796800*z^20 +
2948686820352000*z^24 - 40649991782400*z^28 +  277762867200*z^32 -
920125440*z^36 + 1048576*z^40

where Q[m,n] is a Generalized Okamoto Polynomial, another rational
solution to P_IV. A plot of the roots displays an even more interesting
structure, with a quasi-rectangular core and 4 triangular "lobes"
(related, in some way, to the the Yablonskii­Vorob¹ev Polynomials, Q[n],
which are Rational Solutions of P_II)

If you overlay plots of the roots of H[4,4] and Q[4,4] what is most
striking is that the "common" roots are very similar. In other words,
H[4,4] is "approximately" a factor of Q[4,4].

Is there a "standard" way of quantifying this approximate factorization?

This approximate factorization holds for any m and n and appears to
improve as m and n increases. Perhaps there is a limit in which the
factorization holds asymptotically?

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)
AUSTRALIA                               http://physics.uwa.edu.au/~paul

```

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