On 8 August 2014 03:21, rjf <[email protected]> wrote:
> There is a different answer from Maxima, at least different from what is
> posted.
>
> algebraic:true;
> tellrat(x^3+3*x+1);
> resultant(f,g,y)
> gives
>
> 2201*x^2-2042*x-885
This certainly the correct answer, as one can check by going back to
first principles: the resultant of 2 quadratics is a homogenous
polynomial in the 6 coefficients, actually bihomogeneous of bidegree
(2,2), given by a 4x4 determinant. It makes no difference when one
reduces modulo the modulus:
sage: x = polygen(QQ)
sage: a,b,c,d,e,f = [x^4, x^3+1, x-2, x^3+2*x+1, x+1, x^4+x^3+x^2+1]
sage: M = Matrix(4,4,[a,0,d,0,b,a,e,d,c,b,f,e,0,c,0,f]); M
[ x^4 0 x^3 + 2*x + 1
0]
[ x^3 + 1 x^4 x + 1 x^3
+ 2*x + 1]
[ x - 2 x^3 + 1 x^4 + x^3 + x^2 + 1
x + 1]
[ 0 x - 2 0 x^4 + x^3
+ x^2 + 1]
sage: M.det() % (x^3+3*x+1)
2201*x^2 - 2042*x - 885
John
>
> Note that Maxima can also compute the resultant wrt x without
> renaming variables.
>
>
>
> On Thursday, August 7, 2014 10:25:19 AM UTC-7, Bill Hart wrote:
>>
>> Thanks Peter. That clears it up well.
>>
>> I actually thought that you could specify which variable Pari/GP took the
>> resultant with respect to. But this obviously doesn't get you around the
>> variable ordering issue.
>>
>> I was vaguely aware of that problem, but had no idea I wasn't working
>> around it already.
>>
>> I'm glad to hear this is also fixed in Sage. I tend to use the latest
>> stable release of Sage, but not usually the beta, so I missed that this had
>> been fixed.
>>
>> Thanks again for the help.
>>
>> Bill.
>>
>> On Thursday, 7 August 2014 17:02:56 UTC+2, Peter Bruin wrote:
>>>
>>> Hi Bill,
>>>
>>> This bug was only fixed two weeks ago (Trac tickets #15061 and #16360),
>>> so it only works in the most recent beta versions of Sage.
>>>
>>> The reason why PARI gives the wrong answer has to do with variable
>>> ordering; for example, f is translated to
>>>
>>> f = Mod(-3*y^2*x^2 + (-y^2 - 3*y + 1)*x - 2, x^3 + 3*x + 1)
>>>
>>> which is not a polynomial at all. If you want to do this computation in
>>> PARI, you have to replace x by a variable with lower priority than y:
>>>
>>> gp > y; z;
>>> gp > f = (z^4*y^2 + (z^3 + 1)*y + (z - 2)) * Mod(1, z^3 + 3*z + 1) *
>>> Mod(1, 487326487);
>>> gp > g = ((z^3 + 2*z + 1)*y^2 + (z + 1)*y + (z^4 + z^3 + z^2 + 1)) *
>>> Mod(1, z^3 + 3*z + 1) * Mod(1, 487326487);
>>> gp > polresultant(f, g)
>>> %4 = Mod(Mod(2201, 487326487)*z^2 + Mod(487324445, 487326487)*z +
>>> Mod(487325602, 487326487), z^3 + 3*z + 1)
>>>
>>> Peter
>>>
>>>
>>> Bill Hart wrote:
>>>
>>> > I think that is the correct answer. It agrees with one of the answers
>>> > Magma gives, and someone else says Mathematica agrees.
>>> > My version of Sage is 'Sage Version 6.2, Release Date: 2014-05-06'
>>> > Bill.
>>> >
>>> > On Thursday, 7 August 2014 16:10:52 UTC+2, Frédéric Chapoton wrote:
>>> >
>>> > Hello,
>>> > it works for me on sage 6.3.beta8, giving:
>>> > 2201*xbar^2 + 487324445*xbar + 487325602
>>> > which version of sage do you use ? type version() to know that.
>>> > By the way, you should rather have asked that question on
>>> > sage-support
>>> > or on ask.sagemath.org
>>> > Le jeudi 7 août 2014 16:03:43 UTC+2, Bill Hart a écrit :
>>> >
>>> > I'm having difficulties with the polynomial resultant function in
>>> > Sage.
>>> > I'm trying to find the resultant of two polynomials.
>>> > I compute the resultant of the polynomials below to be:
>>> > 6768454*x^2+257200062*x+20305258
>>> > Pari/GP says the resultant is 1, which I don't believe, so I thought
>>> > I'd try Sage 6.2:
>>> > sage: R = Integers(487326487)
>>> > sage: S.<x> = PolynomialRing(R)
>>> > sage: T = QuotientRing(S, x^3 + 3*x + 1)
>>> > sage: U.<y> = PolynomialRing(T)
>>> > sage: f = x^4*y^2 + (x^3 + 1)*y + (x - 2)
>>> > sage: g = (x^3 + 2*x + 1)*y^2 + (x + 1)*y + (x^4 + x^3 + x^2 + 1)
>>> > sage: f.resultant(g)
>>> >
>>> > ---------------------------------------------------------------------------
>>> > PariError Traceback (most recent call
>>> > last)
>>> > <ipython-input-9-14f40f55d982> in <module>()
>>> > ----> 1 f.resultant(g)
>>> >
>>> > /usr/local/sage/sage-current/local/lib/python2.7/site-packages/sage/structure/element.so
>>> > in sage.structure.element.NamedBinopMethod.__call__
>>> > (sage/structure/element.c:25475)()
>>> >
>>> > /usr/local/sage/sage-current/local/lib/python2.7/site-packages/sage/rings/polynomial/polynomial_element.so
>>> > in sage.rings.polynomial.polynomial_element.Polynomial.resultant
>>> > (sage/rings/polynomial/polynomial_element.c:33908)()
>>> >
>>> > /usr/local/sage/sage-current/local/lib/python2.7/site-packages/sage/rings/polynomial/polynomial_element.so
>>> > in sage.rings.polynomial.polynomial_element.Polynomial._pari_with_name
>>> > (sage/rings/polynomial/polynomial_element.c:33119)()
>>> >
>>> > /usr/local/sage/sage-current/local/lib/python2.7/site-packages/sage/libs/pari/gen.so
>>> > in sage.libs.pari.gen.gen.Polrev (sage/libs/pari/gen.c:12936)()
>>> >
>>> > /usr/local/sage/sage-current/local/lib/python2.7/site-packages/sage/libs/pari/handle_error.so
>>> > in sage.libs.pari.handle_error._pari_handle_exception
>>> > (sage/libs/pari/handle_error.c:1178)()
>>> > PariError: variable must have higher priority in gtopoly
>>> > -------------------
>>> > Magma gives two different answers, depending on whether you coerce the
>>> > coefficients of f and g into T or not (I don't see why). Neither agree
>>> > with the Pari/GP answer, or the answer I compute myself (which is very
>>> > likely wrong).
>>> > Coercing the coefficients into T makes no difference to the resulting
>>> > error in Sage.
>>> > Can someone tell me what I am doing wrong here. I obviously don't know
>>> > what I'm doing. I've never really tried to do any genuine computer
>>> > algebra before.
>>> > Bill.
>>> >
>>> > --
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>>>
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