On Thursday, February 16, 2017 at 7:46:18 AM UTC, Dima Pasechnik wrote:
>
>
>
> On Thursday, February 16, 2017 at 6:59:04 AM UTC, William wrote:
>>
>> **Disclaimer: I consider myself very naive about computational 
>> commutative algebra, especially with floating point numbers.  Dima, 
>> thanks for answering the question, but I think you are maybe jumping 
>> to wronc conclusions. See below. ** 
>>
>> > The backend that actually does this computation is Singular, adn it 
>> does not do floating point numbers (in this context for sure). 
>>
>> It's possible you are completely wrong.  
>
>
> well, not 100% :-)
> I maintain that it's well-known how to get examples of Groebner bases of 
> ideas which will not be 
> representable with original precision---if you work in RR(n), for some n, 
> say.
> I don't know what Singular does, but it cannot always work in limited 
> precision, full stop.
>

here you can see an example of Singular's Groebner basis computation 
returning a wrong result for 
polynomials with (low precision) floating point coeffcients:
 
https://trac.sagemath.org/ticket/22387#comment:2 

(and Sage 7.4 unable to make sense of this Singular output, but that's 
another story)
 

> They also "do" Groebner bases for monomial orders for which termination is 
> not guaranteed...
>
>  
>
>> Half the intro examples in 
>> the "make a ring" section of the Singular manual are floating point: 
>> https://www.singular.uni-kl.de/Manual/4-0-3/sing_29.htm 
>>
>> ~$ singular 
>>                      SINGULAR                                 / 
>>  A Computer Algebra System for Polynomial Computations       /   version 
>> 4.1.0 
>>                                                            0< 
>>  by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann     \   Nov 2016 
>> FB Mathematik der Universitaet, D-67653 Kaiserslautern        \ 
>> > ring r = (real,50),(x,y,z),dp; 
>> > poly f=x3+yx2+3y+4; 
>> > qring q=std(maxideal(2)); 
>> > basering; 
>> //   characteristic : 0 (real:50 digits, additional 50 digits) 
>> //   number of vars : 3 
>> //        block   1 : ordering dp 
>> //                  : names    x y z 
>> //        block   2 : ordering C 
>> // quotient ring from ideal 
>> _[1]=z2 
>> _[2]=yz 
>> _[3]=xz 
>> _[4]=y2 
>> _[5]=xy 
>> _[6]=x2 
>> > poly g=fetch(r, f); 
>> > g; 
>> x3+x2y+3*y+4 
>> > 4.5*g; 
>> 4.5*x3+4.5*x2y+13.5*y+18 
>> > reduce(g,std(0)); 
>> 3*y+4 
>> > 
>>
>> > I have opened https://trac.sagemath.org/ticket/22387 to deal with 
>> this. 
>>
>> Please fix this, e.g., the ticket should point out that Sage appeared 
>> to work and provide results before. 
>>
>> Just because you think it's nuts to use floating point numbers in the 
>> context of commutative algebra doesn't mean it is...   It's a whole 
>> research area, e.g., 
>>
>> http://link.springer.com/chapter/10.1007/978-3-540-87827-8_23 
>>
>
> a paper from 2008 with one citation and one self-citation is not a "whole 
> research area", come on...
>  
>
>>
>> It's even possible the Singular developers aren't naive. 
>>
>  
> well, they are not - although they allow the users to shoot themselves in 
> all the feet around them.
>   
>
>>
>> **Disclaimer: I consider myself very naive about computational 
>> commutative algebra, especially with floating point numbers.** 
>>
>> William 
>>
>> > 
>> > On Thursday, February 16, 2017 at 5:32:42 AM UTC, Matthew Macauley 
>> wrote: 
>> >> 
>> >> Typing the following: 
>> >> 
>> >> 
>> >> P.<x,y,z> = PolynomialRing(RR, 3, order='lex'); P 
>> >> I = ideal(x^2+y^2+z^2-1, x^2-y+z^2, x-z); I 
>> >> B = I.groebner_basis(); B 
>> >> 
>> >> 
>> >> gives a brutal error (type it into SageMathCell and you'll see). 
>> >> 
>> >> 
>> >> Seth Sullivant suggested that it's due to a roundoff error, because it 
>> >> works with fields such as "QQ" or "GF(3)", etc. That said, I am 99% 
>> sure 
>> >> that it's a relatively new error, because I have typed in those exact 
>> lines 
>> >> in previous semesters (it's from a HW problem that I assigned) and I 
>> haven't 
>> >> had any prior issues. 
>> > 
>> > -- 
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>>
>>
>> -- 
>> William (http://wstein.org) 
>>
>

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