Find polynomials in S, then find basis for ideal (S)

[rapid1]

Hi There,

I posted this question over at MHF to no avail, I'm not really sure what the ruling is on this kind of thing, I know this site was setup when MHF was down for a long time but you seem to still be active and a lot of clever people are still here so hopefully you don't mind taking a look at this for me :)

I have a couple of example questions that I'm trying to get my head around, a bit of guidance would be fabulous.\(S:=\{f\in\mathcal{Q}[X,Y]\mid f(X,Y)=f(Y,X) \mbox{ and } \deg(f)\geq 0\}\)1a: Give two polynomials that belong to \(S\).
1b: Find a finite basis of the ideal \((S)\) of \(\mathcal{Q}[X,Y]\) and justify your answer.I then have the question where the questions are the same but based on this
\(S:=\{f\in\mathcal{Q}[X,Y]\mid f(X,Y)=-f(Y,X)\}.\)

 

[HallsofIvy]

Hi There,

I posted this question over at MHF to no avail, I'm not really sure what the ruling is on this kind of thing, I know this site was setup when MHF was down for a long time but you seem to still be active and a lot of clever people are still here so hopefully you don't mind taking a look at this for me :)

I have a couple of example questions that I'm trying to get my head around, a bit of guidance would be fabulous.\(S:=\{f\in\mathcal{Q}[X,Y]\mid f(X,Y)=f(Y,X) \mbox{ and } \deg(f)\geq 0\}\)1a: Give two polynomials that belong to \(S\).

Do you understand what Q[X,Y] is? It is the set of all polynomials in variables X and Y with rational coefficients. Examples are X+ Y, $$3X^2+ 2Y$$, and [tex]X^2+ XY+ Y^2[tex]
To be in S requires that it be symmetric- that is that swapping X and Y does not change the polynomial. X+ Y and $$X^2+ XY+ Y^2$$ are in S but $$3X^2+ 2Y$$ is not.

1b: Find a finite basis of the ideal \((S)\) of \(\mathcal{Q}[X,Y]\) and justify your answer.I then have the question where the questions are the same but based on this
\(S:=\{f\in\mathcal{Q}[X,Y]\mid f(X,Y)=-f(Y,X)\}.\)

 

[rapid1]

Yeh, I thought that would be the case, thanks for confirming. What about part b however, a finite basis?

Also with the second question where \(f(X,Y)=-f(Y,X)\) I'm honestly struggling to think of any polynomials, other than \(0\), that fit because the minus makes it more tricky.

 

[Opalg]

Also with the second question where \(f(X,Y)=-f(Y,X)\) I'm honestly struggling to think of any polynomials, other than \(0\), that fit because the minus makes it more tricky.

How about $X-Y$ ?

 

[blue_raver22]

1a: Two polynomials that belong to S are \(f(X,Y)=X^2+Y^2\) and \(g(X,Y)=X^3+Y^3\).

1b: A finite basis for the ideal \((S)\) can be given by the set \(\{X^2+Y^2, X^3+Y^3\}\). This set spans the ideal because any polynomial \(f\in S\) can be written as a linear combination of these two polynomials (since any polynomial in \(S\) must have terms of even degree and terms of odd degree). This set is also linearly independent, as there is no non-zero linear combination of these two polynomials that will give the zero polynomial. Therefore, this set forms a basis for the ideal \((S)\).

For the second question, a similar approach can be taken. A finite basis for the ideal \((S)\) can be given by the set \(\{X^2, XY, Y^2\}\). This set also spans the ideal because any polynomial \(f\in S\) can be written as a linear combination of these three polynomials. This set is also linearly independent, as there is no non-zero linear combination of these three polynomials that will give the zero polynomial. Therefore, this set forms a basis for the ideal \((S)\).