To compute the grobner basis of two polynomials the command is well known: R.ideal(x^2+y^2-1, (x+y)^2-1).groebner_basis(), where the ring is defined as R.<x,y> = PolynomialRing(QQ). I want to compute this for n polynomials $f_1, f_2,\ldots,f_n$ where $ n$ is as my choice, so how to use the loop for to solve this problem
This can be done exactly the same way for an arbitrary number of polynomials.
You didn't specify the number of variables ; I suppose you want as many variables as polynomials.
An example, using three polynomials of three variables :
sage: R1=PolynomialRing(QQ, "t", 3)
sage: S1=[R1.random_element() for u in range(3)] ; S1
[3/119*t1^2 - 3*t2^2 + 1/2*t0 + 5*t2,
-t0^2 - 1/21*t2^2 - t0 - 2*t1 + 1/3,
1/3*t0*t1 - t2^2 + t0]
sage: J1=R1.ideal(S1)
sage: B1=J1.groebner_basis() ; B1
[t2^4 - 85/132*t2^3 - 23258975/34848*t0*t2 - 595/22*t1*t2 - 46929213143/1463616*t2^2 + 2596929205/487872*t0 - 10833061/17424*t1 + 306569585/5808*t2 + 23211719/209088, t0*t2^2 - 85/132*t0*t2 + 273635/5544*t2^2 - 14929/1848*t0 + 85/66*t1 - 1785/22*t2 - 179/792, t1*t2^2 + 1785/44*t0*t2 + 171997/88*t2^2 - 28517/88*t0 + 833/22*t1 - 70805/22*t2 - 595/88, t0^2 + 1/21*t2^2 + t0 + 2*t1 - 1/3, t0*t1 - 3*t2^2 + 3*t0, t1^2 - 119*t2^2 + 119/6*t0 + 595/3*t2]
Writing a function generalizing to an arbitrary number of polynomials/variables is lazily left as an exercise to the reader.
HTH,
Asked: 2024-06-14 22:57:27 +0100
Seen: 130 times
Last updated: Jun 15
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