BUCHBERGER ALGORITHM  •  GRÖBNER BASIS  •  SPARSE MULTIVARIATE POLYNOMIALS

Define p1, p2,...,pm by

pk = ak1xbk1 + ak2xbk2 + ak3xbk3 + ... + aknxbkn
where This is merely a set of m multivariate polynomials. We calculate a Gröbner Basis for the Ideal I = <p1,p2,...pm> in Q[x1,x2,...xr]. In entering the polynomials, terms and factors must be entered in strict lexicographic order (i.e. x1 comes first!)
To input I = <p1,p2,...pm> use
  • 1. a vertical line | to separate polynomials
  • 2. a semicolon ; to separate terms within a polynomial
  • 3. commas to separate factors in a term
2. and 3. must be entered in lexicographic order. For 3. the sequence is [numerator of coefficient],[denominator of coefficient],[index (label) of 1st. variable],[exponent of 1st. variable],[index (label) of 2nd. variable],[exponent of 2nd. variable], etc....

For example to input

x2z + (1/2)x + y2
-4x2 - xy2 + xz - (1/4)
2x + y2z + (1/2)

enter  

1,1,1,2,3,1;1,2,1,1;1,1,2,2|-4,1,1,2;-1,1,1,1,2,2;1,1,1,1,3,1;-1,4|2,1,1,1;1,1,2,2,3,1;1,2
Enter generating polynomials for I:
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Michael Mc Gettrick