The singular points on f = x^2 y - x y on a plane

[naturemath]

Let f(x,y) = x^2 y - xy = x(x-1)y be a polynomial in k[x,y].

I am looking for the singular subset of this function.

Taking the partials, we obtain

f_x = 2xy - y

f_y = x^2 - x.

In order to find the singular subset, both partials (with respect to x and with respect to y) must vanish. So we obtain that

f_x = 2xy - y = 0

which implies x = 1/2 or y = 0,

while

f_y = x^2 - x = 0

implies

x = 0 or x =1.

Drawing a picture of f, it is clear that the two points (0,0) and (1,0) are singular points, but what does x = 1/2 tell us? Is this point supposed to be a singular point as well?

 

[haruspex]

f_y = x^2 - x = 0
:
what does x = 1/2 tell us? Is this point supposed to be a singular point as well?

No. f_y is never zero at x = 1/2.

 

[Millennial]

Applying this method generates the POSSIBLE singular points. You should check if they really are singular points by other means.

 

[naturemath]

Thank you haruspex and Millennial. That clarifies things!

 

[HallsofIvy]

A some what better method would be to look at f_x= 2xy- y= (2x- 1)y= 0 and say, as you do, that either x= 1/2 or y= 0.

Now look at f_y= x^2- x= x(x- 1)= 0 and use the values you already have. IF x= 1/2, that is impossible but IF y= 0, x can be 0 or 1. The singular points are (0, 0) and (1, 0).

 

[naturemath]

Ah, that's a good argument. Thanks HallsofIvy!