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

Thank you haruspex and Millennial. That clarifies things!

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...

Thanks Fredrik! Your explanation clarifies a lot.

Oh I see. Thank you mathwonk!

Thanks micromass. This notation is commonly found in books on quantum mechanics. So are physicists just complicating a notion that supposed to be simple? I'm not attacking physicists but I feel that there must be more/other reasons why they use such notation...

I've always wondered about this notation: <ej|A|ei>. I understand the pairing involved in <A ei,ej> but what does this <ej|A|ei> mean?

> This is hardly a proof. I'm thinking of writing the ideal (or any ideal) as a primary decomposition, but for even that, is there a systematic way to decompose an ideal in such a way? Or do you recommend other (more) feasible options?

> Yes, but strictly speaking you'll need to show that neither x1 nor (x2 x4-x3^2) is in the ideal. Thanks. > This is hardly a proof. Yes, but it seems quite messy to do it directly, using the polynomials.

So (f1,f2)R/(x1,x2)R is generated by x5 x7 and x6 x8, right? I.e., (f1,f2)R/(x1,x2)R = (x5 x7, x6 x8)

So I'm guessing the following. Q1. Is this (x1, x2) a prime ideal in C[x1, x2, x3, x4] ? Yes since the quotient is an integral domain (an irredu variety-- it's the x3 x4-plane). Q2. What about this: (x1 x4-x2 x3, x1 x3-x2^2)? No since x2 (x1 x4-x2 x3) -x3 (x1 x3-x2^2) = x1(x2 x4-x3^2)...

This is a basic abstract algebra question. Q1. Is this (x1, x2) a prime ideal in C[x1, x2, x3, x4] ? Q2. What about this: (x1 x4-x2 x3, x1 x3-x22)? Q3. Is this a prime ideal (this is the twisted cubic in projective 3-space): (x1 x4-x2 x3, x1 x3-x22, x2 x4-x32)? Thanks everyone.

It seems to me that in (f1,f2)R/(x1,x2)R, we are killing off all polynomial combination of x1 and x2 (all polys of the form ax1 + bx2 where a and b are in R), or are we killing off something smaller than that? Thus, do \bar{f}1 = x5 x7 \bar{f}2 = x6 x8 in (f1,f2)R/(x1,x2)R ?

So what I'm interested in is: how do the images of f1 and f2 look like in (f1,f2)R/(x1,x2)R?

> (f1,f2)R/(x1,x2)R? The numerator (f1, f2) is an ideal in R generated by f1 and f2 while the denominator (x1, x2) is an ideal in R generated by x1 and x2.