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If one has a one variable polynomial then the discriminant can be used to test whether the polynomial has any repeated roots or equivalently where the polynomial and its derivative have a repeated root. Now If I look at a hyper-surface (Let us say the zero set of some polynomial equation $F(x_1,\cdots,x_n)=0$) then are there some polynomials in the coefficients of $F$ which tells me whether the variety contains singular points?

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Alternatively, what is any effective way of testing whether a surface has a singular point without looking for a point on the variety where the gradient vanishes? – CPM Jul 19 '11 at 12:39
The singular locus is the variety cut out by the ideal generated by $F$ and its partial derivatives, and in general there's no reason to expect this ideal to be principal. – Qiaochu Yuan Jul 19 '11 at 12:47
As Qiaochu Yuan explained there is no single polynomial that would tell you, whether the defining polynomial and all its partial derivatives have a common zero. An algorithmic way of deciding this would be to use Gröbner basis techniques. I don't know, if any computer algebra system has this available. When facing a task like this I usually try ad hoc techniques, but I don't have too much experience with that. – Jyrki Lahtonen Jul 19 '11 at 14:59
@Qiaochu, Yes, good point. Do you know how I would find explicit polynomials for the singular locus in terms of the coefficients of the original polynomial? – CPM Jul 19 '11 at 15:51
@CPM: are you looking for something more specific than "$F$ and its partial derivatives"? – Qiaochu Yuan Jul 19 '11 at 16:02
up vote 7 down vote accepted

Let $F$ be a homogenous polynomial. There is a polynomial $\Delta$, in the coefficients of $F$, which vanishes precisely when the projective hypersurface $F=0$ is singular. This polynomial is called the "$A$-discriminant", a term which is unfortunately impossible to google for.

Most references on this are going to want to study the case of a hypersurface in a general toric variety, which they will encode by a finite set $A$ of lattice points. To help get you oriented: Classical homogenous polynomials correspond to projective space. If $F$ is homogenous of degree $d$ in $n$ variables, then the set $A$ is $$\left\{ (a_1, \ldots, a_n) \in \mathbb{Z}^n : a_i \geq 0 \ \mbox{and} \ \sum a_i = d \right\}.$$

Unfortunately, I don't know a simple description of the $A$-discriminant to give you, and I know that computing them is difficult enough that it is used as a benchmark for computer algebra systems. I think testing whether the ideal generated by $F$ and its partial derivatives is irrelevant (using, for example, Macaulay II) should be much easier than computing the corresponding $A$-discriminant. But I am not an expert on the computational practicalities.

The standard reference is

I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky: Discriminants, Resultants, and Multidimensional Determinants; Birkauser, Boston, MA, 1994.

If anyone knows a briefer, more accessible reference, please edit this answer or leave a comment.

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