specialization theorems

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nob...@nowhere.invalid

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Mar 13, 2020, 5:07:56 PM3/13/20
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Still nothing official is to be found on the internet of Masser and
Zannier's work that lead to counterexamples to a theorem by James H.
Davenport: Some parametrized algebraic functions not identically
integrable can actually be integrated for infinitely many parameter
values.

Nothing official, that is, apart from David Masser's feature article
"Integration in elementary terms" in the LMS Newsletter (Newsletter
London Math. Soc. 473 (2017), 30-36), made available here:

<https://webusers.imj-prg.fr/~jan.nekovar/co/ter/int.pdf>

But there is a very nice survey article by Umberto Zannier, entitled
"Some specialization theorems for families of abelian varieties", which
has appeared (or is to appear) in the Münster Journal of Mathematics:

<http://wwwmath1.uni-muenster.de/mjm/acc/mjm-Zannier.pdf>

Martin.

nob...@nowhere.invalid

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Sep 10, 2020, 1:20:10 PM9/10/20
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"clicl...@freenet.de" schrieb:
>
> Still nothing official is to be found on the internet of Masser and
> Zannier's work that lead to counterexamples to a theorem by James H.
> Davenport: Some parametrized algebraic functions not identically
> integrable can actually be integrated for infinitely many parameter
> values.
>
> Nothing official, that is, apart from David Masser's feature article
> "Integration in elementary terms" in the LMS Newsletter (Newsletter
> London Math. Soc. 473 (2017), 30-36), made available here:
>
> <https://webusers.imj-prg.fr/~jan.nekovar/co/ter/int.pdf>
>
> [...]
>

Just noticed that an official preprint has become available at:

<https://dmi.unibas.ch/de/personen/david-masser/publikationen/>

One quick remark: The antiderivatives of integrals (14.4) and (21.11)
could be simplified substantially: their numerators and denominators in
essence are squares. As written, the logarithms look more impressive
though! ... Hmmm.

Martin.

Валерий Заподовников

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Oct 26, 2021, 3:37:44 AM10/26/21
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Валерий Заподовников

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Feb 5, 2022, 7:01:24 PMFeb 5
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вторник, 26 октября 2021 г. в 10:37:44 UTC+3, Валерий Заподовников:
Can we discuss this paper? I mean they refuted Davenport's theorem
there, modified it to be true and even gave insane amount of different
integrals, including some criticism/typos of Greenhill book! Very nice.

And looks like it has some implications on Risch algorithm.

Валерий Заподовников

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Jun 9, 2022, 9:32:39 PMJun 9
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To dev of FriCAS: apparently FriCAS cannot handle the key
result of both papers, it causes infinite loop (or whatever):

integrate(x/(x^2-1/5-2*%i/5)/(x^3-x)^(1/2), x)

While it is elementary! Mathematica 13 can do it very fast, but
not in elementary functions. It is interesting WHY FullSimplify
does not see it from the math. 13 result, is it possible there is
a simplification to elementary function possible or a constant
difference in real part? Or is the result in the paper too big?
__________
The other example that Mathematica 13 solves with insanely
big result. Never seen anything like it (but again paper gives
elementary result, DID not check FriCAS):

Integrate[((5t^2+40^t+62)x+t^3+8t^2+70^t+144)/ (x−t)((2t+8)x+t^2+4t+18)( x^3−30x−56)^(1/2),x]

P.S. After reading the papers I did not find the script to
generate those but it should be there, of course.

nob...@nowhere.invalid

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Jun 10, 2022, 6:38:52 AMJun 10
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??????? ???????????? schrieb:
>
> To dev of FriCAS: apparently FriCAS cannot handle the key
> result of both papers, it causes infinite loop (or whatever):
>
> integrate(x/(x^2-1/5-2*%i/5)/(x^3-x)^(1/2), x)

If I remember correctly, Waldek Hebisch found that FriCAS could handle
this integral a few years ago, after some bugs had been eliminated.

>
> While it is elementary! Mathematica 13 can do it very fast, but
> not in elementary functions. It is interesting WHY FullSimplify
> does not see it from the math. 13 result, is it possible there is
> a simplification to elementary function possible or a constant
> difference in real part? Or is the result in the paper too big?

I suspect you are asking simply too much of Mathematica here.

> __________
> The other example that Mathematica 13 solves with insanely
> big result. Never seen anything like it (but again paper gives
> elementary result, DID not check FriCAS):
>
> Integrate[((5t^2+40^t+62)x+t^3+8t^2+70^t+144)/ (x-t)((2t+8)x+t^2+4t+18)( x^3-30x-56)^(1/2),x]
>
> P.S. After reading the papers I did not find the script to
> generate those but it should be there, of course.

[In the above, I have corrected your minus signs to ASCII.] In the
published paper, which you can find at the Acta Mathematica site for
download, this counterexample was withdrawn; the authors finally
managed to show that the antiderivative is elementary for at most 138
values of t (see the end of Section 16.3). What value did you try?

Martin.

Валерий Заподовников

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Jun 10, 2022, 10:18:39 AMJun 10
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Did you actually check? How much time do you need to solve it?

10 hours?

>I suspect you are asking simply too much of Mathematica here.

But it does try IntegrateAlgebraic, but fails...

Yep, the second is not a counterexample, not elementary
(page 301, they discuss how FriCAS helped them see it is
not elementary). Mathematica agrees and gives crazy result.

Okay.

Валерий Заподовников

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Jun 10, 2022, 12:16:55 PMJun 10
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Wait a second, it is you who is Detmar Martin Welz in the paper.

Then do you know where is this code? "Partly computationally" 138 t of 21.12, what are they?

>In fact, we were able to show, partly computationally, that Q(i√2) does not turn up, and we

Oh and also I do not have the old preprint with wrong proofs, can you give it?

BTW, the "sceptical" part is just nice, can you maybe also check what did you wrote to them?
And what did you do with FriCAS?

There is also a numerical integration "proof", which they obviously copied from 2017 paper:

>which Maple 18 cannot check even by differentiation (however it can check equality up to say
1000 decimal places when we integrate between say x = 2 and x = 2.1).

I think that is illegal. You cannot just check small part from 2 to 21/10. Did not do FullSimplify,
too lazy :), the other Maple 18 "comment" can be now done thanks to IntegrateAlgebraic:

SetSystemOptions[
"IntegrateOptions" -> {"IntegrateAlgebraicTimeConstraint" -> 100}];

Integrate[(5 x - 1)/Sqrt[x^4 + 2 x^2 - 4 x + 1], x]

It would nice of them to prove like here https://hdl.handle.net/2346/45299
Different kind of paper, I suppose.

nob...@nowhere.invalid

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Jun 11, 2022, 2:12:09 AMJun 11
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??????? ???????????? schrieb:
With respect to:

integrate(x/(x^2 - 1/5 - 2*%i/5)/sqrt(x^3 - x), x)

I can confirm that the current online version of FriCAS runs into a
timeout. Ten hours would be ridiculously long for this computation,
however. Perhaps FriCAS went stale and a fresh version is needed.

The integral on p.232 of the paper:

integrate((5*x - 1)/sqrt(x^4 + 2*x^2 - 4*x + 1), x)

is an Abel case; you may consult Gunter & Kuzmin's "Sbornik zadach po
vysshei matematike" for a brief treatment and other examples (p.52-53
of vol.2 at <https://techlibrary.ru/bookpage.htm>). Derive 6.10 returns
the present example unevaluated whereas FriCAS 1.3.7 solves it right
away.

Otherwise, your queries about contents, background, or genesis of the
paper must be addressed to the authors.

Martin.

Валерий Заподовников

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Jun 11, 2022, 4:46:50 AMJun 11
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So you do not consider yourself one of the authors? :)

>I can confirm that the current online version of FriCAS

What is version you have in your linux? I use the latest, since
I use Debian testing. Will try 10 hours after all. Maupybe will
compile first HEAD of master.

>The integral on p.232 of the paper:

Well, that was just a Maple 18 attack example in the paper, I
do not care about THAT one. After all, this is just Zolotarev
integral (solved by Chebyshev before, proved by him):
https://en.wikipedia.org/wiki/Yegor_Ivanovich_Zolotarev

Abel did even come close to solving common case, LOL.

An example also right on "Risch algorithm" wikipedia page.

nob...@nowhere.invalid

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Jun 13, 2022, 2:45:01 AM (13 days ago) Jun 13
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??????? ???????????? schrieb:
>
> What is version you have in your linux? I use the latest, since
> I use Debian testing. Will try 10 hours after all. Maupybe will
> compile first HEAD of master.
>

I was referring to the web-interface at

<https://fricas-wiki.math.uni.wroc.pl/FriCASIntegration#bottom>

where FriCAS identifies as version 1.3.7. Don't worry - Waldek Hebisch
should have become aware of the problem with this integrand by now. If
something broke, he can probably fix it. Extended computations failing
late into their execution can take a lot of time to debug, though.

Martin.

Валерий Заподовников

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Jun 13, 2022, 3:16:42 AM (13 days ago) Jun 13
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I did check 1.3.7 in my debian, it also timeouts after 8 hours
and maybe even crashes, since it exits as if )quit happened.

nob...@nowhere.invalid

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Jun 14, 2022, 2:12:36 AM (12 days ago) Jun 14
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??????? ???????????? schrieb:
>
> I did check 1.3.7 in my debian, it also timeouts after 8 hours
> and maybe even crashes, since it exits as if )quit happened.

I have dug into the old sci.math.symbolic posts. On February 11, 2017
in the thread "alarum: Risch integrator fails to divide by zero",
Waldek Hebisch indeed reported:

>
> Using division polynomials I have found points of order 5,
> one corresponds to u^2 = (1 + 2*sqrt(-1))/5. FriCAS have
> now computed integrals for orders 3, 5, 6, 8. The results
> are rather lengthy, so instead of posting them here I have
> put them at:
>
> http://www.math.uni.wroc.pl/~hebisch/fricas/p3
>
> (3 above means order 3, replace 3 by 5, 6, 8 for higher order).
>

And his FriCAS antiderivative of this order-5 Masser-Zannier integrand
has actually survived at:

<https://www.math.uni.wroc.pl/~hebisch/fricas/p5>

The coefficients of 20-plus digits are rather intimidating. This must
have been computed with version 1.3.0 or 1.3.1, presumably using some
hot fixes. So either something broke in the meantime, or FriCAS does
not read your %i as sqrt(-1). But it still knows that %i*%i equals -1.
I think it may really have gone stale. A fresh version of FriCAS should
be supplied!

Martin.

Валерий Заподовников

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Jun 15, 2022, 8:53:31 AM (10 days ago) Jun 15
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Well, apparently the author reads this indeed:

https://github.com/fricas/fricas/commit/1f42999f91ce516a8d027a61be4ecbf32ad2ada4

of course this is very strange:

testIntegrate("(5*x-1)/sqrt(x^4 + 2*x^2 - 4*x + 1)", "x", "alg")

since it is really Zolotarev case, in fact same stuff is in
https://en.wikipedia.org/wiki/Risch_algorithm

From the paper (p. 230): testIntegrate("5*x^2/sqrt(x^6 +x)", "x", "alg"),
but it is not that important. (Also "Massetr and Zanier", typos.)

>found points of order 5,
> one corresponds to u^2 = (1 + 2*sqrt(-1))/5.

Yes, that is in the paper too, page 233. In fact t = (1/5(5-10i))^2 is indeed
(1 + 2*sqrt(-1))/5.

Both those examples work even before the commit.

I did compile HEAD of master today, nothing helps our case though.
Will compile 1.3.2 and check it out. Will not do bisect though, hope
author will find the regression commit.

nob...@nowhere.invalid

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Jun 16, 2022, 2:17:34 AM (10 days ago) Jun 16
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??????? ???????????? schrieb:
>
> Well, apparently the author reads this indeed:
>
> https://github.com/fricas/fricas/commit/1f42999f91ce516a8d027a61be4ecbf32ad2ada4
>
> of course this is very strange:
>
> testIntegrate("(5*x-1)/sqrt(x^4 + 2*x^2 - 4*x + 1)", "x", "alg")
>
> since it is really Zolotarev case, in fact same stuff is in
> https://en.wikipedia.org/wiki/Risch_algorithm

Since this integral resolves into a logarithm of an algebraic function,
my earlier reference to Gunter & Kuzmin was inappropriate as it deals
with purely algebraic antidervatives. Both cases were treated by Abel
separately.

>
> From the paper (p. 230): testIntegrate("5*x^2/sqrt(x^6 +x)", "x", "alg"),

integrate(5*x^2/sqrt(x^6 + x), x) works on the web interface running
version 1.3.7, the instantaneous result being:

log(2*x^2*(x^6+x)^(1/2)+(2*x^5+1)).

> but it is not that important. (Also "Massetr and Zanier", typos.)
>
> >found points of order 5,
> > one corresponds to u^2 = (1 + 2*sqrt(-1))/5.
>
> Yes, that is in the paper too, page 233. In fact t = (1/5(5-10i))^2
> is indeed (1 + 2*sqrt(-1))/5.
>
> Both those examples work even before the commit.
>
> I did compile HEAD of master today, nothing helps our case though.
> Will compile 1.3.2 and check it out. Will not do bisect though, hope
> author will find the regression commit.

I am pretty sure the FriCAS developers are reading your messages on
this newsgroup. But they could have other priorities.

A more direct venue for problem reports is the fricas-devel newsgroup
which is moderated and local to Google Groups.

Martin.

Валерий Заподовников

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Jun 16, 2022, 8:23:22 AM (10 days ago) Jun 16
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Case closed, you can do it all with

setSimplifyDenomsFlag(true)

integrate(x/((x^2 - ((1 + 2*sqrt(-1))/5))*sqrt(x^3 - x)), x)

Apparently it does not like %i, since that
https://github.com/fricas/fricas/pull/92#issuecomment-1157581265

nob...@nowhere.invalid

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Jun 16, 2022, 9:24:34 AM (9 days ago) Jun 16
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??????? ???????????? schrieb:
Glad to have made you happy. In view of the disastrous consequences, the
developers should insert the automatic substitution of %i by sqrt(-1) at
the start of the FriCAS integrator.

Martin.
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