DistributedMultivariatePolynomial which is abbreviated as DMP and HomogeneousDistributedMultivariatePolynomial, which is abbreviated as HDMP, are very similar to MultivariatePolynomial except that they are represented and displayed in a non-recursive manner.
(d1,d2,d3) : DMP([z,y,x],FRAC INT)
Type: Void
The constructor DMP orders its monomials lexicographically while HDMP orders them by total order refined by reverse lexicographic order.
d1 := -4*z + 4*y**2*x + 16*x**2 + 1
2 2
- 4z + 4y x + 16x + 1
Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer)
d2 := 2*z*y**2 + 4*x + 1
2
2z y + 4x + 1
Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer)
d3 := 2*z*x**2 - 2*y**2 - x
2 2
2z x - 2y - x
Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer)
These constructors are mostly used in Groebner basis calculations.
groebner [d1,d2,d3]
1568 6 1264 5 6 4 182 3 2047 2 103 2857
[z - ---- x - ---- x + --- x + --- x - ---- x - ---- x - -----,
2745 305 305 549 610 2745 10980
2 112 6 84 5 1264 4 13 3 84 2 1772 2
y + ---- x - --- x - ---- x - --- x + --- x + ---- x + ----,
2745 305 305 549 305 2745 2745
7 29 6 17 4 11 3 1 2 15 1
x + -- x - -- x - -- x + -- x + -- x + -]
4 16 8 32 16 4
Type: List DistributedMultivariatePolynomial([z,y,x],Fraction Integer)
(n1,n2,n3) : HDMP([z,y,x],FRAC INT)
Type: Void
n1 := d1
2 2
4y x + 16x - 4z + 1
Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer)
n2 := d2
2
2z y + 4x + 1
Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer)
n3 := d3
2 2
2z x - 2y - x
Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer)
Note that we get a different Groebner basis when we use the HDMP polynomials, as expected.
groebner [n1,n2,n3]
4 3 3 2 1 1 4 29 3 1 2 7 9 1
[y + 2x - - x + - z - -, x + -- x - - y - - z x - -- x - -,
2 2 8 4 8 4 16 4
2 1 2 2 1 2 2 1
z y + 2x + -, y x + 4x - z + -, z x - y - - x,
2 4 2
2 2 2 1 3
z - 4y + 2x - - z - - x]
4 2
Type: List HomogeneousDistributedMultivariatePolynomial([z,y,x],
Fraction Integer)
GeneralDistributedMultivariatePolynomial is somewhat more flexible in the sense that as well as accepting a list of variables to specify the variable ordering, it also takes a predicate on exponent vectors to specify the term ordering. With this polynomial type the user can experiment with the effect of using completely arbitrary term orderings. This flexibility is mostly important for algorithms such as Groebner basis calculations which can be very sensitive to term ordering.
See Also: