I am wondering if there is a software or algorithm out there that helps to compute the dimension and generators of a quotient of a polynomial ring with coefficients on a field $K$ (usually rationals or real coefficients).
For example, let $R= K[x,y]$ and $J$ be the ideal generated by $xy- x^3, x^4 + y^2 + 1, y^3- x^2$. Then $R/J$ is generated as a $K$-vector space by $1, x, x^2, x^3, y, y^2$.
I am having a polynomial ring in many more variables and my ideals are generated by way more complicated polynomials. I am not very into software like Maple, Sage, Mathematica and I don't know if there is a tool there to solve my problem.
A good CAS is the software Macaulay2.