# Software computing dimension and degree

Assume a projective scheme $$X_{k_1,\dots,k_r}\subset\mathbb{P}^n$$ is given as the set of common solutions of homogeneous polynomials $$F_1(x_0,\dots,x_n),\dots,F_s(x_0,\dots,x_n)$$, where the $$F_i$$ depends on parameters $$k_1,\dots,k_r$$ varying in the base field.

Does there exist a computer algebra software that can compute the dimension and the degree of $$X_{k_1,\dots,k_r}$$ as functions of $$k_1,\dots,k_r$$?

• I think dimension and degree follow, if a grobner base is known. So the problem is to compute all grobner bases appearing for certain polynomial equalities and inequalities in the paramers $k_1,\ldots,k_r$. I did not verify this by doing an example computation myself, but singular.uni-kl.de/Manual/4-0-3/sing_955.htm and singular.uni-kl.de/Manual/4-0-3/sing_931.htm#SEC1006 seem to provide, what you sought. (Libraries compregb_lib and grobcov_lib). Dec 30 '20 at 13:53

## 1 Answer

In addition to Magma (tagged in your question), Singular (cited in Jürgen Böhm's comment), there's also Macaulay2. AFAIK, all provide methods for calculating the required Gröbner basis.

• Could you give a reference for Macaulay2 of a documentation page where computing gröbner bases with parameters is described? Note that this is an extension of the usual concept of gröbner base: If you have $R=\mathbb{Q}[a][x]$ as basering with parameter $a$ and a polynom $f = a x^2 + x + 1$ then a gröbner base with parameters would give a list $[a \neq 0, f]$ as a first base and $[a=0, x+1]$ as a second base. This is useful in many applications, but I have not found a command in Macaulay2 to do this. The same applies to your Magma link. Jan 13 '21 at 17:18