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 X^2+ XY+ Y^2<br /> 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.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> 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<br /> \(S:=\{f\in\mathcal{Q}[X,Y]\mid f(X,Y)=-f(Y,X)\}.\) </div> </div> </blockquote>

 

[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 $X-Y$ ?