Multivariate Monomials

**MONOMIAL**
is a Python 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^nwe 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^3The 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

- x1 is less than x2;
- or x1 = x2, but y1 is less than y2;
- or x1 = x2, and y1 = y2, but z1 is less than z2;

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:

- z1 is greater than z2;
- or z1 = z2 but y1 is greater than y2;
- or z1 = z2, and y1 = y2, but x1 is greater than x2.

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 FORTRAN77 version and
a FORTRAN90 version and
a MATLAB version and
a Python version.

LEGENDRE_PRODUCT_POLYNOMIAL, a Python library which defines Legendre product polynomials, creating a multivariate polynomial as the product of univariate Legendre polynomials.

POLYNOMIAL, a Python library which adds, multiplies, differentiates, evaluates and prints multivariate polynomials in a space of M dimensions.

SUBSET, a Python library which enumerates, generates, ranks and unranks combinatorial objects including combinations, compositions, Gray codes, index sets, partitions, permutations, subsets, and Young tables.

- i4_choose.py, computes the binomial coefficient C(N,K).
- i4_uniform_ab.py, returns a scaled pseudorandom I4 between A and B.
- i4vec_print.py, prints an I4VEC.
- i4vec_sum.py, returns the sum of the entries of an I4VEC.
- i4vec_uniform_ab.py, returns a scaled pseudorandom I4VEC between A and B.
- mono_between_enum.py, enumerates monomials in D dimensions of degrees in a range.
- mono_between_next_grevlex.py, next monomial in grevlex order, total degree between N1 and N2.
- mono_between_next_grlex.py, next monomial in grlex order, total degree between N1 and N2.
- mono_between_random.py, returns a random monomial with total degree between N1 and N2.
- mono_next_grevlex.py, next monomial in grevlex order.
- mono_next_grlex.py, next monomial in grlex order.
- mono_print.py, prints a monomial.
- mono_rank_grlex.py, ranks a monomial in grlex order.
- mono_total_enum.py, enumerates monomials in D dimensions of degree equal to N.
- mono_total_next_grevlex.py, next monomial in grevlex order, total degree equals N.
- mono_total_next_grlex.py, next monomial in grlex order, total degree equals N.
- mono_total_random.py, random monomial with total degree N.
- mono_unrank_grlex.py, unranks a monomial in grlex order.
- mono_upto_enum.py, enumerates monomials in D dimensions of degree between 0 and N.
- mono_upto_next_grevlex.py, next monomial in grevlex order, total degree up to N.
- mono_upto_next_grlex.py, next monomial in grlex order, total degree up to N.
- mono_upto_random.py, random monomial with total degree up to N.
- mono_value.py, evaluates a monomial.
- timestamp.py, prints the current YMDHMS date as a time stamp.

- monomial_test.py, a sample calling program.
- monomial_test.sh, BASH commands to compile and run the sample program.
- monomial_test_output.txt, the output file.

You can go up one level to the Python source codes.