Singular

If you want to reduce the polynomials in the variables a_0_ etc
then use [u]5.1.111 reduce[/u]
http://www.singular.uni-kl.de/Manual/latest/sing_277.htm#SEC318

reduce(I,J); generalizes the polynomial division, where you put in
J the polynomials which should do the reduction.

Example:
[code]
> ring R=0,a,dp;
> poly f = 2a4+a2+1;
> poly g = a3+2a2+1;
> reduce(f,g); // It gives the remainder by polynomial division
// ** _ is no standard basis // This is only a warning
9a2-2a+5
> reduce(f,std(g));
9a2-2a+5
> division (f,g);
[1]:
_[1,1]=2a-4
[2]:
_[1]=9a2-2a+5
[3]:
_[1,1]=1
> f == (2a-4)*g + 9a2-2a+5; // representation of f= q*g +r
1
[/code]

Yes, it is right for X's degree.
The problem is for other varaibles. Will they be reduced by division algorithm?

For example, will "a_0_0_^4" be reduced by division algorithm?

Thanks a lot.
Gepo

[quote="gepo"]When we compute GB over GF(2^k), if root of minpoly becomes greater than k, the modulo operation will be triggered.
My doubt is whether this happens on other variable rather than root of minpoly?

For example,
ring r=(2,X),(a0,b0,a1,b1,a2,b2,a3,b3,
a_0_0_,b_0_0_,a_0_1_,b_0_1_,
a_1_0_,b_1_0_,a_1_1_,b_1_1_,
A0,B0,A1,B1,
AA,BB,CC),lp;
minpoly=1+X^3+X^4;

I want to know what happens when all the variables' degrees become greater than 4 (except "X"'s degree)?
Is there a modulo operation triggered?
Like "a_0_0_^4" will be reduced by "minpoly=1+X^3+X^4"?

By the way, is there a command to dump all the results of internal procedures, like results of s-poly, reducetion procedure?
[/quote]

The minpoly reduces only the expressions in X, and this reduction
is polynomial division. The resulting [b]remainder[/b] of polynomial
division by X^4+X^3+1 is a polynomial of degree less than 4.

Thus, if the input has degree less than 4 the output will be the input.

There is no [i]trigger[/i] but the mechanism is always applied.

Did you see:
[u]5.1.22 dump[/u]
http://www.singular.uni-kl.de/Manual/latest/sing_188.htm#SEC229

When we compute GB over GF(2^k), if root of minpoly becomes greater than k, the modulo operation will be triggered.
My doubt is whether this happens on other variable rather than root of minpoly?

For example,

ring r=(2,X),(a0,b0,a1,b1,a2,b2,a3,b3,
a_0_0_,b_0_0_,a_0_1_,b_0_1_,
a_1_0_,b_1_0_,a_1_1_,b_1_1_,
A0,B0,A1,B1,
AA,BB,CC),lp;

minpoly=1+X^3+X^4;
ideal I=
a_0_0_+a0+a3,
a_0_1_+a2+a3,
a_1_0_+a1+a3,
a_1_1_+a2,
b_0_0_+b0+b3,
b_0_1_+b2+b3,
b_1_0_+b1+b3,
b_1_1_+b2,
A0+a_0_0_+a_0_1_*(X^5),
A1+a_1_0_+a_1_1_*(X^5),
B0+b_0_0_+b_0_1_*(X^5),
B1+b_1_0_+b_1_1_*(X^5),
AA+A0*X^0+A1*X^1,
BB+B0*X^0+B1*X^1,
CC+AA*BB,;
groebner(I);

I want to know what happens when all the variables' degrees become greater than 4 (except "X"'s degree)?
Is there a modulo operation triggered?
Like "a_0_0_^4" will be reduced by "minpoly=1+X^3+X^4"?


By the way, is there a command to dump all the results of internal procedures, like results of s-poly, reducetion procedure?
Thanks a lot
Gepo