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

#### rapid

##### New member
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

##### Well-known member
MHB Math Helper
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 $$X^2+ XY+ Y^2[tex] To be in S requires that it be symmetric- that is that swapping X and Y does not change the polynomial. X+ Y and [tex]X^2+ XY+ Y^2$$ 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
$$S:=\{f\in\mathcal{Q}[X,Y]\mid f(X,Y)=-f(Y,X)\}.$$

#### rapid

##### New member
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

##### MHB Oldtimer
Staff member
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 $X-Y$ ?