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Re: Re: Minimum input for GroebnerBasis
A couple of minor corrections.
> What is true is that the set {x + y + 1, x - y} and the set {x + y +
> 1, 2 x + 2 y + 2, x - y} generate the same ideal (or if you prefer not
> to mention ideals you can say that they have the corresponding systems
> of equations ahve the same set of solutions).
I am not saying that the concept of an ideal of a polynomial ring and
the set of zeros of its generators are equivalent; the relationship is
more subtle. But you can get a fair idea about what it is all about if
you think of an ideal in terms of the set of common zeros of its
generators (or, of course, all its elements).
> In general, for a given monomial order a Groebner basis can have any
> finite number of elements,
Of course I meant "arbitrarily large" number of elements. It can't
have fewer elements than a minimal Groebner basis for the given
ordering.
Andrzej Kozlowski
On 8 Jul 2008, at 20:48, Andrzej Kozlowski wrote:
> Your question seems to be based on a misunderstanding of what a
> Groebner basis is. A GrobenerBasis is a basis (i.e. a se t of
> generators) for the ideal generated by a set of polynomials and in
> general is not, a subset of the set of polynomials that you started
> from. (If you do not know what an ideal is it will be hard to
> understand the rest of this so I suggest you look up any textbook of
> modern abstract algebra or even simply polynomial algebra.) Only in
> rare cases a subset of the original set of generators is a Groebner
> basis. (Of course there is a trivial way to produce such cases, which
> I describe at the end).
>
> Next, turning to your example: the claim that Q is a Groebner basis
> for the ideal generated by P is false (at least for the default
> Lexicographic Order). A Groebner basis for an ideal is defined a set
> of generators of the ideal with the property that, for any element of
> the ideal, it's leading term (with respect to the given monomial
> order!) is divisible by the leading term of some element of the basis.
> So consider the ideal generated by
>
> {x + y + 1, 2 x + 2 y + 2, x - y}
>
> Now, the element (x + y + 1)-(x-y) = 2y+1 is in the ideal. It's
> leading term is 2y. Now consider the set Q = {x + y + 1, x - y}. With
> the lexicographic ordering (x comes before y) the leading terms of
> both elements of Q are x. But 2y is not divisible by x. So Q is not a
> GrobenerBasis for this ordering. (It also does not satisfy the so
> called "Buchberger criterion"). On the other hand, {1 + 2 y, 1 + 2 x}
> is a Grobner basis and 2y is divisible by the leading term of 1+2y,
> which is 2y.
>
> What is true is that the set {x + y + 1, x - y} and the set {x + y +
> 1, 2 x + 2 y + 2, x - y} generate the same ideal (or if you prefer not
> to mention ideals you can say that they have the corresponding systems
> of equations ahve the same set of solutions). In other words, the set
> {x + y + 1, x - y} is a basis for the ideal generated by {x + y + 1, 2
> x + 2 y + 2, x - y}. A basis, yes, but a Groebner basis, no. This is
> the main point
>
> Next, a few more remarks of a more general nature, which may be
> related to your question. A Groebner basis is defined only for a given
> monomial order. There is no such thing as "the Groebner basis" there
> is only a Groebner basis for some specific order. If you look at the
> options MonomialOrder in Mathematica's GroebnerBasis function you will
> see its default value.
>
> In[3]:= Options[GroebnerBasis, MonomialOrder]
> Out[3]= {MonomialOrder -> Lexicographic}
>
> the Lexicographic order is the default. It is not necessarily the best
> order for all purposes (but it is the most suited for using
> elimination).
>
> You can check that in many cases the number of elements in a Groebner
> basis for an ideal with respect to one order will be different form
> the number of elements in a Groebner basis for the same ideal with
> respect to a different monomial order. For example:
>
> Length[GroebnerBasis[{x y + z - x z, x^2 - z, 2 x^3 - x^2 y z - 1},
> {x, y, z},
> MonomialOrder -> Lexicographic]]
> 3
>
> Length[GroebnerBasis[{x y + z - x z, x^2 - z, 2 x^3 - x^2 y z - 1},
> {x, y, z},
> MonomialOrder -> DegreeLexicographic]]
> 4
>
>
> In general, for a given monomial order a Groebner basis can have any
> finite number of elements, since once you have a Groebner basis for an
> ideal, you can just adjoin to it any element of the ideal and it will
> still be a Groebner basis. But, if you require that a Groebner basis
> be a reduced one, that is, have the property that for any two
> polynomials in the basis no monomial appearing in one of them is a
> multiple of the leading term of the other than such a basis will be
> minimal - that is, if you remove any polynomial from it it will no
> longer be a Groebner basis (with respect to the given monomial order
> of course). In fact, such a reduced Groebner basis is unique if you
> add the requirement that the leading coefficient of each element in
> the Groebner basis is 1. Most computer algebra systems (including
> Mathematica) return a reduced Groebner basis, but not necessarily a
> monic one. So the Groebner basis returned by Mathematica is already a
> minimal one, in other words it has the property that if you remove
> anything from it it will no longer be a Groebner basis.
>
>
> Andrzej Kozlowski
>
>
> On 8 Jul 2008, at 15:25, TuesdayShopping wrote:
>
>> Given a finite set of polynomials P in variables belonging to V, we
>> compute the GroebnerBasis G. What is the set of polynomials Q (Q is
>> a subset of P), such that (a) Q produces the same GroebnerBasis G;
>> and, (b) if any element from the set Q is removed, Q will no longer
>> produce G. In other words, Q is the minimum set of polynomials (from
>> P) required, in order to produce the Groebner Basis G. If there are
>> several Q's possible, we will want the one with the smallest number
>> of polynomials in in it. If there are several with the same number,
>> will want the first one we encounter. Question is how do we find Q
>> in Mathematica?
>>
>> For example, P can be {x + y + 1, 2 x + 2 y + 2, x - y}
>>
>> GroebnerBasis[{x + y + 1, 2 x + 2 y + 2, x - y}, {x,y}] is {1 + 2 y,
>> 1 + 2 x}
>>
>> Q is {x + y + 1, x - y}
>>
>
>
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