# Primary decomposition with parameters

$$\newcommand\QQ{\mathbb{Q}}$$

Considering the polynomial $$f = x^2 - a y$$ one notes, that it is irreducible in $$\QQ[x,y]$$ for all $$a \neq 0 \in \QQ$$ and factors for $$a = 0$$.

More generally, let $$A=k[a]$$ with a computable field $$k$$ and $$B=A[x_1,\ldots,x_n]$$ together with an ideal $$I=(f_1,\ldots,f_r) \subseteq B$$.

Now, how could one compute a polynomial $$g(a) \in A$$, such that

• for all $$a_0 \in \bar{k}$$ (the algebraic closure of $$k$$) with $$g(a_0) \neq 0$$ the ideal $$(I, a - a_0)$$ has primary decomposition of "the same type" in $$R = \bar{k}[x_1,\ldots,x_n]$$ (corresponding to $$x^2 - a y$$ irreducible for $$a \neq 0$$ in the starting example) and

• for $$g(a_0) = 0$$ the ideal $$(I, a - a_0)$$ has primary decomposition of an "exceptional type" in $$R$$ (corresponding to reducibility for $$a = 0$$ in the example).

It seems to me that there is no prepackaged solution for this in Macaulay2 (which I mostly use for computations in algebraic geometry).

Looking into Singular, I found out, that it can calculate "comprehensive grobner bases", which means in this concrete example it can calculate polynomials $$g_1(a),\ldots,g_s(a)$$, such that for $$g_i(a_0) \neq 0$$ the ideal $$(I, a - a_0)$$ has a groebner base in $$B$$ of a form which is unique for $$g_i(a_0) \neq 0$$ and differs fom the other $$i$$.

For the example in the beginning one gets $$g_1(a) = a, g_2(a)=1$$ with associated bases $$(y a - x^2)$$ and $$(a, x^2)$$ respectively. But this works only for a termorder where $$y a$$ is the lead monomial in $$x^2 - a y$$.

# Questions:

1) How to compute such a $$g(a)$$ mentioned above, classifying primary decomposition types into "ordinary" and "exceptional"?

2) Can it be done with a suitable form of "comprehensive grobner base" calculation?

3) Can it be done with a combination of other "standard" calculations available, say in Macaulay2?

4) What about generalizations where $$A=k[a_1,\ldots,a_p]$$ has more than one parameter?