Strong and weak conjectures on essential specializations and the discriminant ideal

About 20 years old · traced to 2006

Let $I$ be the ideal under consideration, and let $G$ be its generic Gröbner basis. A specialization is essential when its reduced Gröbner basis is not the generic basis $G$ , and singular when it is a singular specialization. Let $J$ and $N$ be the ideals defined in the paper, with $N$ the relevant intersection of the radical ideals $N_i$ .

Strong and weak conjectures. The strong conjecture asserts that all essential specializations are singular. The weak conjecture asserts

J\supseteq N.

The paper proves the reverse inclusion $J\subseteq N$ , so the weak conjecture would imply $J=N$ . The supplied text does not give evidence that either conjecture has been resolved.

References

Primary source

Montserrat Manubens and Antonio Montes, “Improving DISPGB Algorithm Using the Discriminant Ideal”, arXiv:math/0601763 (2006).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Open

A reader-written construction claims both conjectures are false, but no independent verification was found.

The strong conjecture says every essential specialization is singular; the weak conjecture says one discriminant ideal contains the other. The supplied formulation records only the reverse inclusion and no established resolution.

Posted attempt

A reader-written construction claims a counterexample to both conjectures. It uses an ideal over a two-parameter polynomial ring and asserts an essential but nonsingular specialization, together with strict containment of the two ideals. The attempt is presented as a complete disproof, but it has not been independently verified.

Current status (as of August 2026): a complete counterexample to both conjectures has been claimed, but neither conjecture is verified as false and the matter remains open.

CounterexampleThis solution needs a summarySee full solutionHide full solution

Both conjectures are false, even when the singular-specialization locus is nonempty and both discriminant ideals are proper.

The construction amplifies the one-parameter example in M. Wibmer, Gröbner bases for families of affine or projective schemes, Journal of Symbolic Computation 42 (2007), Example 3, https://arxiv.org/abs/math/0608019. A related example was also recorded by A. Montes, On the canonical discussion of polynomial systems with parameters, https://arxiv.org/abs/math/0601674.

Let

R=\mathbb Q[u,v]

and consider

I=\langle vu(ux-1),\,vx(ux-1)\rangle =v(ux-1)\langle u,x\rangle \subseteq R[x].

Over $\mathbb Q(u,v)$ , the generic reduced Gröbner basis is

G=\{g\},\qquad g=x-\frac1u,

with leading-power-product set $\{x\}$ .

For $(u,v)=(c,b)$ , the specialized ideal is

I_{c,b}= \begin{cases} \langle x-1/c\rangle,&b\neq0,\ c\neq0,\\ \langle x\rangle,&b\neq0,\ c=0,\\ (0),&b=0. \end{cases}

Under the original definition, a specialization is singular precisely when its leading-power-product set differs from the generic set. Therefore the singular locus is exactly $V(v)$ , and hence

N=I(V(v))=(v). \tag{1}

On the other hand,

J_g=\{a\in R:ag\in I\}.

If $ag\in R[x]$ , its constant coefficient $-a/u$ forces $a=uh$ for some $h\in R$ . Cancelling $ux-1$ in the integral domain $R[x]$ gives

ag=h(ux-1)\in v(ux-1)(u,x) \quad\Longleftrightarrow\quad h\in v(u,x)\cap R=(uv).

Thus

J_g=(u^2v), \qquad J=\sqrt{J_g}=(uv). \tag{2}

At the point $(u,v)=(0,1)$ ,

J_g\subseteq(u,v-1),

so the specialization is essential. Nevertheless its specialized ideal is $\langle x\rangle$ , whose leading-power-product set is still $\{x\}$ ; therefore it is not singular. This disproves the strong conjecture.

Finally, (1) and (2) give

J=(uv)\subsetneq(v)=N,

so $v\in N\setminus J$ . Hence the conjectured inclusion $J\supseteq N$ also fails, disproving the weak conjecture.