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Changing Monomial Order


Changing Monomial Order






Often one wishes to change the monomial order of an ideal.  Magma allows
one to do this by use of the ChangeOrder function.



ChangeOrder(I, Q) : RngMPol, RngMPol -> RngMPol, Map



Given an ideal I of the polynomial ring P = R[x_1, ..., x_n],
together with a polynomial ring Q of rank n (with possibly a
different order to that of P),
return the ideal J of Q corresponding to I and the isomorphism f
from P to Q.  The map f simply maps P.i to Q.i for each i.
The point of the function is that one can change the order on monomials
of I to be that of Q.  When a Gröbner basis of J is needed
to be calculated, Magma uses the Gröbner Walk starting from a
Gröbner basis of I if possible---this usually makes order conversion
much more efficient than computing a Gröbner basis of J from scratch.




ChangeOrder(I, order) : RngMPol, ..., -> RngMPol, Map



Given an ideal I of the polynomial ring P = R[x_1, ..., x_n],
together with a monomial order order, construct the polynomial ring
Q = R[x_1, ..., x_n] with order order, and then
return the ideal J of Q corresponding to I and the isomorphism f
from P to Q.  See the section on monomial orders for the valid
values for the argument order.
The map f simply maps P.i to Q.i for each i.
The point of the function is that one can change the order on monomials
of I to be order.  When a Gröbner basis of J is needed
to be calculated, Magma uses the Gröbner Walk starting from a
Gröbner basis of I if possible---this usually makes order conversion
much more efficient than computing a Gröbner basis of J from scratch.





Example RngMPol_ChangeOrder (H29E21)

We write a function univgen which, given a zero-dimensional ideal,
computes the univariate elimination ideal generator
for a particular variable by changing order to the appropriate
univariate order.  Note that this function is the same as (and is
in fact implemented in exactly the same way as) the intrinsic function
UnivariateEliminationIdealGenerator.  We then find the appropriate
univariate polynomials for a particular ideal.





> function univgen(I, i)
>    // Make sure I has a Groebner basis so that
>    // the Walk algorithm will be used when
>    // constructing a Groebner basis of J
>    Groebner(I);
>    J := ChangeOrder(I, "univ", i);
>    Groebner(J);
>    return rep{f: f in Basis(J) | IsUnivariate(f, i)};
> end function;
>
> P<x, y, z> := PolynomialRing(RationalField(), 3, "grevlex");
> I := ideal<P |
>     1 - x + x*y^2 - x*z^2,
>     1 - y + y*x^2 + y*z^2,
>     1 - z - z*x^2 + z*y^2 >;
>
> univgen(I, 1);
x^21 - x^20 - 2*x^19 + 4*x^18 - 5/2*x^17 - 5/2*x^16 + 4*x^15 - 
    15/2*x^14 + 129/16*x^13 + 11/16*x^12 - 103/8*x^11 + 
    131/8*x^10 - 49/16*x^9 - 171/16*x^8 + 12*x^7 - 3*x^6 - 
    29/8*x^5 + 15/4*x^4 - 17/16*x^3 - 5/16*x^2 + 5/16*x - 1/16
> univgen(I, 2);
y^14 - y^13 - 13/2*y^12 + 8*y^11 + 53/4*y^10 - 97/4*y^9 - 
    45/8*y^8 + 33*y^7 - 25/2*y^6 - 18*y^5 + 107/8*y^4 + 5/8*y^3 -
    27/8*y^2 + 9/8*y - 1/8
> univgen(I, 3);
z^21 - z^20 - 2*z^19 + 4*z^18 - 5/2*z^17 - 5/2*z^16 + 4*z^15 - 
    15/2*z^14 + 129/16*z^13 + 11/16*z^12 - 103/8*z^11 + 
    131/8*z^10 - 49/16*z^9 - 171/16*z^8 + 12*z^7 - 3*z^6 - 
    29/8*z^5 + 15/4*z^4 - 17/16*z^3 - 5/16*z^2 + 5/16*z - 1/16





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