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Re: 136 theorems on 29 pages
Posted:
Oct 7, 2012 2:48 PM
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clicliclic@freenet.de wrote:
>
> Waldek Hebisch schrieb:
> >
> > clicliclic@freenet.de wrote:
> > >
> > > And citing inaccessible theses and
> > > internal reports can be more frustrating than helpful to a reader
> > > (as well as dangerous to an author who never saw them).
> >
> > Just a quck comment: IME theses tend to be freely publically avaliable
> > and frequently give more detailed exposition than journal papers,
> > so actually can be _more_ helpful than other puplications (which
> > frequently are harder to obtain).
> >
> > AFAICS Rotstein thesis is available at:
> >
> > www.cs.kent.edu/~rothstei/dis.pdf
> >
> > (second Google hit for Rotstein "Aspects of symbolic").
> > Bronstein thesis was published in Journal of Symbolic Computation.
> > In the past access was restricted, but it seems that now
> > anybody can go to
> >
> > http://www.sciencedirect.com/science/journal/07477171/9/2
> >
> > (Issue 2 of Volume 9) and choose second article.
> >
> > It seems that in Trager case some amout of Googling should
> > produce downloadable copy.
> >
>
> 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.
--
Waldek Hebisch
hebisch@math.uni.wroc.pl
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