Abhinav Baid wrote:
>
> On 07/18/2015 12:08 AM, Waldek Hebisch wrote:
> > Abhinav Baid wrote:
> >> On Wednesday, July 15, 2015 at 8:23:45 AM UTC+5:30, Waldek Hebisch wrote:
> >>> 4) In 'compute_bound' you seem to ignore ramified exponential
> >>> parts. This looks wrong.
> >>>
> >> Sorry, I don't get what I'm missing here. First, I check that the degree of
> >>
> >> 'ram' = 1 and then only the constant coefficient of 'expart' matters in
> >> further computation. So, I think it should work?
> > When 'ram' = 1 this looks OK. But AFAICS ramified singularity
> > still may have generalized exponent contributing to constant term.
> > As an example consider operator in the example in the section
> > 2.7. The operator has only ramified generalized exponents, but
> > the residue is nonzero. So in Lemma 27 in van Hoej thesis you
> > need to take into account all generalized exponents.
> Okay. Is the current code fine, then?
There is problem with computing trace. If is better to avoid
going via AlgebraicNumber here (in general we would like to
allow non-algebraic constants here) and the formula you use
look wrong. A simple routine to comput trace may look like:
get_trace(f : F, k : K) : F ==
min_pol := minPoly(k)
Sae := SimpleAlgebraicExtension(F, UP, min_pol)
fa := univariate(f, k, min_pol)
trace(reduce(fa)$Sae)$Sae
Note it computes trace with respect to one kernel passed
as the argument k. In general you need to call 'kernels'
to get list of kernels and compute trace for each algebraic
one. If there are nonalgebraic kernels you need to replace
them by rational numbers (almost all values are OK, you
only need to avoid division by 0).
Note: you need to do replacement consistently for all
singularities.
Also, van Hoej has "v'(e_i - e_j)" term in his formula. I
see nothing corresponding to this in your code.
> >>> 5) Infinity is used differently in bounds: other singularities
> >>> bound denomiantors of a_i. Infinity bounds differences
> >>> between degrees of mumerators and denominators of a_i (so
> >>> given bound on denominators infinity gives bound for degree
> >>> of numarators).
> >>>
> >>
> >> Again, I apologize because I think I don't understand this. What are the
> >> a_i in question?
> > We are trying to find a factor R = \sum_i a_i\partial_x^i
> >
> > so a_i are (unknown) coefficients of the factor. To prove
> > that operator is irreducible (which is one of possible results
> > of van Hoej algorithm) we need bounds on degree of denominators
> > and numerators of a_i. Only having such bounds we can conclude
> > that no solution to Hermite-Pade problem means that operator
> > is irreducible.
> >
> Oh, I get the meaning of a_i now. But, in section 3.6, we are trying to
> find a_i in k[x] and so, there'll be no x in the denominator. So, is
> this about some other computation that I'm (not) doing in another
> function?
We can write operator in monic form:
L = \partial_x + r
with rational r, or we can clear denomonators and write:
L = a_1\partial_x + a_0
with polynomial a_1 and a_0. In other places van Hoej assumes
monic operators, in particular bound are done for monic operators.
But in section 3.6 the Hermite-Pade part is done with a_i in k[x].
So a_m represents lcm of denominators of coefficients of monic
form.
BTW, I think you should now export function to compute generalized
exponents. One reason is that you need generalized exponents
in factor_global, so it makes sense to have functions that
computes them and returns in convenient form. Another reason
is that parts dealing with Chapter 2 of van Hoej thesis are
done now, so it would be good to do more testing. Function
giving generalized exponents would help in testing.
And of course generalized exponents are useful for other
purposes.
--
Waldek Hebisch
[email protected]
--
You received this message because you are subscribed to the Google Groups
"FriCAS - computer algebra system" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/fricas-devel.
For more options, visit https://groups.google.com/d/optout.
