• specialization theorems

    From nobody@nowhere.invalid@21:1/5 to clicliclic@freenet.de on Thu Sep 10 19:23:03 2020
    "clicliclic@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.

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  • From =?UTF-8?B?0JLQsNC70LXRgNC40Lkg0JfQs@21:1/5 to All on Tue Oct 26 00:37:42 2021
    They published it. https://www.intlpress.com/site/pub/pages/journals/items/acta/content/vols/0225/0002/a002/

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  • From =?UTF-8?B?0JLQsNC70LXRgNC40Lkg0JfQs@21:1/5 to All on Sat Feb 5 16:01:22 2022
    вторник, 26 октября 2021 г. в 10:37:44 UTC+3, Валерий Заподовников:
    They published it. https://www.intlpress.com/site/pub/pages/journals/items/acta/content/vols/0225/0002/a002/

    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.

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  • From =?UTF-8?B?0JLQsNC70LXRgNC40Lkg0JfQs@21:1/5 to All on Thu Jun 9 18:32:38 2022
    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.

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  • From nobody@nowhere.invalid@21:1/5 to All on Fri Jun 10 12:38:43 2022
    ??????? ???????????? 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.

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  • From =?UTF-8?B?0JLQsNC70LXRgNC40Lkg0JfQs@21:1/5 to All on Fri Jun 10 07:18:37 2022
    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.

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  • From =?UTF-8?B?0JLQsNC70LXRgNC40Lkg0JfQs@21:1/5 to All on Fri Jun 10 09:16:54 2022
    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.

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  • From nobody@nowhere.invalid@21:1/5 to All on Sat Jun 11 08:11:58 2022
    ??????? ???????????? schrieb:

    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.

    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.

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  • From =?UTF-8?B?0JLQsNC70LXRgNC40Lkg0JfQs@21:1/5 to All on Sat Jun 11 01:46:49 2022
    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.

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  • From nobody@nowhere.invalid@21:1/5 to All on Mon Jun 13 08:44:53 2022
    ??????? ???????????? 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.

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  • From =?UTF-8?B?0JLQsNC70LXRgNC40Lkg0JfQs@21:1/5 to All on Mon Jun 13 00:16:41 2022
    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.

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  • From nobody@nowhere.invalid@21:1/5 to All on Tue Jun 14 08:12:28 2022
    ??????? ???????????? 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.

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  • From =?UTF-8?B?0JLQsNC70LXRgNC40Lkg0JfQs@21:1/5 to All on Wed Jun 15 05:53:30 2022
    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.

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  • From nobody@nowhere.invalid@21:1/5 to All on Thu Jun 16 08:17:35 2022
    ??????? ???????????? 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.

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  • From =?UTF-8?B?0JLQsNC70LXRgNC40Lkg0JfQs@21:1/5 to All on Thu Jun 16 05:23:20 2022
    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

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  • From nobody@nowhere.invalid@21:1/5 to All on Thu Jun 16 15:24:39 2022
    ??????? ???????????? schrieb:

    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

    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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