The binoms command produces binomials from the columns of a matrix. In its simplest form the matrix consists of constants, and the positive constants in each column yield the exponents of one monomial, while the negative constants yield the exponents of the other, difference monomial. More generally, the matrix may consist of polynomials, and the sum of the coefficients of each entry are used to produce an exponent.
This command may produce surprising results in small characteristics. Nevertheless, one shouldn't cringe at using an integer mod p as an exponent; all integers inside a computer are represented mod some number, and this is at least a number we can control.
% binoms binoms <matrix> <ideal> <result> % type a 0 1 2 3 1 1 1 1 % type r a b c d % binoms a r b % type b b-1 ab-1 a2b-1 a3b-1 % homog b c b % res b c 2 0.1.2..3..4.. computation complete after degree 4 % type c.2 -ab+c2 ac a2c+ac2 b-c -a-c -a2-ac-c2 0 1 0 0 0 1 % binoms c.2 r d % type d 0 -b2+ac -b3+a2d
This example is meant to be a tease. Try varying the order, and using an entire Gröbner basis of syzygies.