MONOMIAL is a C++ library which enumerates, lists, ranks, unranks and randomizes multivariate monomials in a space of D dimensions, with total degree less than N, equal to N, or lying within a given range.
A (univariate) monomial in 1 variable x is simply any (nonnegative integer) power of x:
1, x, x^2, x^3, ...
The exponent of x is termed the degree of the monomial.
Since any polynomial p(x) can be written as
p(x) = c(0) * x^0 + c(1) * x^1 + c(2) * x^2 + ... + c(n) * x^n
we may regard the monomials as a natural basis for the space of
polynomials, in which case the coefficients may be regarded as
the coordinates of the polynomial.
A (multivariate) monomial in D variables x(1), x(2), ..., x(d) is a product of the form
x(1)^e(1) * x(2)^e(2) * ... * x(d)^e(d)
where e(1) through e(d) are nonnegative integers. The sum of the
exponents is termed the degree of the monomial.
Any polynomial in D variables can be written as a linear combination of monomials in D variables. The "total degree" of the polynomial is the maximum of the degrees of the monomials that it comprises. For instance, a polynomial in D = 2 variables of total degree 3 might have the form:
p(x,y) = c(0,0) x^0 y^0
+ c(1,0) x^1 y^0 + c(0,1) x^0 y^1
+ c(2,0) x^2 y^0 + c(1,1) x^1 y^1 + c(0,2) x^0 y^2
+ c(3,0) x^3 y^0 + c(2,1) x^2 y^1 + c(1,2) x^1 y^2 + c(0,3) x^0 y^3
The monomials in D variables can be regarded as
a natural basis for the polynomials in D variables.
For multidimensional polynomials, a number of orderings are possible. Two common orderings are "grlex" (graded lexicographic) and "grevlex" (graded reverse lexicographic). Once an ordering is imposed, each monomial in D variables has a rank, and it is possible to ask (and answer!) the following questions:
As mentioned, two common orderings for monomials are "grlex" (graded lexicographic) and "grevlex" (graded reverse lexicographic). The word "graded" in both names indicates that, for both orderings, one monomial is "less" than another if its total degree is less. Thus, for both orderings, xyz^2 is less than y^5 because a monomial of degree 4 is less than a monomial of degree 5.
But what happens when we compare two monomials of the same degree? For the lexicographic ordering, one monomial is less than another if its vector of exponents is lexicographically less. Given two vectors v1=(x1,y1,z1) and v2=(x2,y2,z2), v1 is less than v2 if
Thus, for the grlex ordering, we first order by degree, and then for two monomials of the same degree, we use the lexicographic ordering. Here is how the grlex ordering would arrange monomials in D=3 dimensions.
# monomial expon
-- --------- -----
1 1 0 0 0
2 z 0 0 1
3 y 0 1 0
4 x 1 0 0
5 z^2 0 0 2
6 y z 0 1 1
7 y^2 0 2 0
8 x z 1 0 1
9 x y 1 1 0
10 x^2 2 0 0
11 z^3 0 0 3
12 y z^2 0 1 2
13 y^2z 0 2 1
14 y^3 0 3 0
15 x z^2 1 0 2
16 x y z 1 1 1
17 x y^2 1 2 0
18 x^2 z 2 0 1
19 x^2y 2 1 0
20 x^3 3 0 0
21 z^4 0 0 4
22 y z^3 0 1 3
23 y^2z^2 0 2 2
24 y^3z 0 3 1
25 y^4 0 4 0
26 x z^3 1 0 3
27 x y z^2 1 1 2
28 x y^2z 1 2 1
29 x y^3 1 3 0
30 x^2 z^2 2 0 2
31 x^2y z 2 1 1
32 x^2y^2 2 2 0
33 x^3 z 3 0 1
34 x^3y 3 1 0
35 x^4 4 0 0
36 z^5 0 0 5
... ......... .....
For the reverse lexicographic ordering, given two vectors, v1=(x1,y1,z1) and v2=(x2,y2,z2), v1 is less than v2 if:
Thus, for the grevlex ordering, we first order by degree, and then for two monomials of the same degree, we use the reverse lexicographic ordering. Here is how the grevlex ordering would arrange monomials in D=3 dimensions.
# monomial expon
-- --------- -----
1 1 0 0 0
2 z 0 0 1
3 y 0 1 0
4 x 1 0 0
5 z^2 0 0 2
6 y z 0 1 1
7 x z 1 0 1
8 y^2 0 2 0
9 x y 1 1 0
10 x^2 2 0 0
11 z^3 0 0 3
12 y z^2 0 1 2
13 x z^2 1 0 2
14 y^2z 0 2 1
15 x y z 1 1 1
16 x^2 z 2 0 1
17 y^3 0 3 0
18 x y^2 1 2 0
19 x^2y 2 1 0
20 x^3 3 0 0
21 z^4 0 0 4
22 y z^3 0 1 3
23 x z^3 1 0 3
24 y^2z^2 0 2 2
25 x y z^2 1 1 2
26 x^2 z^2 2 0 2
27 y^3z^1 0 3 1
28 x y^2z 1 2 1
29 x^2y z 2 1 1
30 x^3 z 3 0 1
31 y^4 0 4 0
32 x y^3 1 3 0
33 x^2y^2 2 2 0
34 x^3y 3 1 0
35 x^4 4 0 0
36 z^5 0 0 5
... ......... .....
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
MONOMIAL is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.
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