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

In summary, we are looking for the singular subset of the polynomial function f(x,y) = x^2 y - xy in k[x,y]. To find the singular points, we take the partials with respect to x and y and set them equal to zero. This yields x = 1/2 or y = 0 for f_x and x = 0 or x = 1 for f_y. However, upon further examination, we can see that the point x = 1/2 is not a singular point. The singular points for this function are (0, 0) and (1, 0). A more accurate method would be to consider both partials simultaneously and use the values we already have to determine
  • #1
naturemath
31
0
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?
 
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  • #2
naturemath said:
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.
 
  • #3
Applying this method generates the POSSIBLE singular points. You should check if they really are singular points by other means.
 
  • #4
Thank you haruspex and Millennial. That clarifies things!
 
  • #5
A some what better method would be to look at [itex]f_x= 2xy- y= (2x- 1)y= 0[/itex] and say, as you do, that either x= 1/2 or y= 0.

Now look at [itex]f_y= x^2- x= x(x- 1)= 0[/itex] 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).
 
  • #6
Ah, that's a good argument. Thanks HallsofIvy!
 

1. What is a singular point?

A singular point is a point on a curve or surface where the derivative of the function is not defined or is equal to zero. In other words, it is a point where the function is not smooth and has a sharp turn or corner.

2. How do you find the singular points on a plane?

To find the singular points on a plane, you can take the partial derivatives of the function with respect to x and y, and set them equal to zero. The points where both partial derivatives are equal to zero are the singular points.

3. What is the significance of singular points on a plane?

Singular points can help us understand the behavior of a function and its critical points. They can also indicate where the function has a local maximum or minimum, or where it changes direction.

4. Can a function have more than one singular point?

Yes, a function can have multiple singular points. This can happen when the function has multiple critical points or when there are points where the function is not smooth in more than one direction.

5. How are singular points related to the graph of a function?

Singular points can be seen as points where the graph of a function has a sharp turn or corner. They can also be seen as points where the graph changes direction or has a local maximum or minimum. In some cases, the graph may have holes or discontinuities at the singular points.

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