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Re: 136 theorems on 29 pages
Posted:
Nov 19, 2012 4:55 PM
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Waldek Hebisch schrieb:
>
> clicliclic@freenet.de wrote:
> >
> > Expecting it to be the most promising, I have checked Bronstein's thesis
> > publication of 1990 (57 pages, 2.4MB, barely readable digitization). I
> > can assure our author that this contains no "prior art" concerning an
> > extension of Hermite reduction (p. 132) to multiple non-integer
> > exponents or to a lowering of exponents: the equations agree with those
> > in the Symbolic Integration Tutorial of 1998/2000 and fail for multiple
> > non-integer exponents because the integrated term is again too
> > restricted.
>
> You seem to ignore the following fact: Bronstein uses integral
> basis, which is equivalent to separating "essential" part under
> root and writing rest as polynomial. That is given P^a with
> P squarefree one writes it as P^nP^b where 0\leq b < 1. After
> such splitting one can collect irrationalities into a single term.
> From Bronstein introduction:
>
> : Using only rational techniques, we are able to remove multiple
> : finite poles of the integrand.
>
> The claim is that he can handle _any_ algebraic integrand (in
> fact more general because he allows algebraics depending on
> exponentials or logarithms). And "removing multiple poles"
> means exactly that he can increase exponents of factor of
> denominator as long as they are smaller than -1. He does not
> explicitly state this, but looking at the proof one sees that
> only new factors which can appear in denominator are terms under
> radical, so indeed his reduction process manages to
> increase powers toward - 1 (at cost of somewhat uncontrolled
> factor in the numerator).
>
> Then, on page 146 Bronstain outlines how to remove multiple
> pole at infinity. He gives little datails (and equation
> S_{log2} looks wrong), still, I have checked that following
> his hints works. Now, removing multiple pole at infinity
> really means lowering exponents, if the function at hand
> is a product.
>
Here's how I remember having read Bronstein's thesis and Tutorial (I was
off-line for six weeks, hence this belated reply):
Bronstein needs to introduce a finite-dimensional basis of algebraic
functions in order to handle arbitrary algebraic integrands. Ordinary
Hermite reduction thereby splits into interdependent parallel reductions
since the basis elements will mix under differentiation. In order to
guarantee success, the basis must be "integral", which is hard to
compute in general. (Later he therefore introduced a "lazy" version of
the parallelized reduction, where such a basis is not needed at the
outset but gradually produced by repeated updates in the course of the
reduction.)
As we know, all this must (and therefore does) simplify for the case of
an algebraic product denominator, and Bronstein seems to consider this
special case too simple to be spelled out (I am inclined to share this
view). In his papers, I have seen no instance of Hermite reduction where
the integrated term does recognizably involve the required product
denominator with exponents close to those in the (original or
transformed) integrand denominator. This holds for the rational as well
as algebraic cases.
After all, both the raising and lowering of denominator exponents (and
consequently any combination of these operations involving any number of
exponents) in an algebraic product denominator requires nothing but
solving a system of linear equations (whose particular structure for the
raising of exponents permits the extended Euclidean algorithm to be
used). This is much easier to see directly than to infer via Bronstein.
I consider it the opposite of helpful to refer to his parallelized
Hermite machinery in order to arrive at this fairly trivial fact!
Martin.
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