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Re: 136 theorems on 29 pages
Posted:
Oct 7, 2012 2:48 PM


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 noninteger > exponents or to a lowering of exponents: the equations agree with those > in the Symbolic Integration Tutorial of 1998/2000 and fail for multiple > noninteger 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



