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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