# Two-generated ideal of $k[x,y]$ invariant under an involution

The following question is a natural continuation of this MSE question; further elaboration can be found in it.

I have now edited my current question, according to the two nice comments that I have received:

Let $$k$$ be a field of characteristic zero.

Let $$f: (x,y) \mapsto (p,q)$$ be a $$k$$-algebra endomorphism of $$k[x,y]$$, $$I=\langle p,q \rangle$$ a proper ideal of $$k[x,y]$$ and $$\delta$$ an involution on $$k[x,y]$$, namely, a $$k$$-algebra automorphism of order two.

Assume that $$\delta(I) \subseteq I$$.

I do not mind to further assume that:

(i) $$\delta: (x,y) \mapsto (x,-y)$$ (ii) $$\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in k-\{0\}$$.

Is the following claim true?

Claim: There exist automorphisms $$g,h$$ of $$k[x,y]$$ such that one of $$\{(gfh)(x), (gfh)(y)\}$$ is symmetric or skew-symmetric w.r.t. some involution on $$k[x,y]$$.

An attempt to answer my question:

Write $$p=s_1+k_1$$ and $$q=s_2+k_2$$, with $$s_1,s_2$$ symmetric and $$k_1,k_2$$ skew-symmetric.

$$\delta(\langle p,q \rangle) \subseteq \langle p,q \rangle$$, implies that $$s_1,k_1,s_2,k_2 \in \langle p,q \rangle$$.

Then we can write: (i) $$s_1=Ap+Bq=(A_s+A_k)(s_1+k_1)+(B_s+B_k)(s_2+k_2)$$, for some $$A=A_s+A_k$$, $$B=B_s+B_k$$, with $$A_s,B_s$$ symmetric, $$A_k,B_k$$ skew-symmetric.

By applying $$\delta$$ to (i) we get (i)', and then (i)+(i)' and (i)-(i)' yield: $$s_1=A_s s_1+A_k k_1+B_s s_2+ B_k k_2$$ and $$0=A_s k_1+A_k s_1+B_s k_2+B_k s_2$$.

Similarly for $$k_1,s_2,k_2$$.

However, I am not sure if this helps (hopefully yes).

Perhaps I am missing a relevnt known result that can answer my question?

What about $$C$$*-algebras? What about the first Weyl algebra? Notice that the subalgebra $$k[p,q]$$ is not invariant under $$\delta$$, unless $$p,q$$ are of one of the above three forms of $$(gfh)(x),(gfh)(y)$$.

The one-dimensional case:

Notice that in $$k[x]$$ an analog result holds for the involution $$\delta_c: x \mapsto -x+c$$, $$c \in k$$, namely:

If $$I$$ is an ideal of $$k[x]$$ which is invariant under $$\delta_c$$, then $$I=\langle h \rangle$$, with $$h$$ symmetric or skew-symmetric with respect to $$\delta_c$$.

The proof of this one-dimensional case can be found in the above mentioned MSE question.

This is a slightly similar question (with $$\mathbb{F}_2[x,y]$$ instead of $$k[x,y]$$, $$p,q$$ homogeneous, etc.).

Thank you very much!

• 18 versions in two days. – Gerry Myerson Sep 12 at 22:10
• – Gerry Myerson Sep 12 at 22:15
• Come off it. You were already up to version 13 well before the edits during that hour. – Gerry Myerson Sep 13 at 9:50
• "Minor edits"? 13 added 108 characters in body; 12 added 37 characters in body; 8 deleted 143 characters in body; edited tags; 7 deleted 143 characters in body; edited tags; 6 added 189 characters in body; 5 added 48 characters in body; 4 deleted 74 characters in body; 2 deleted 45 characters in body. – Gerry Myerson Sep 14 at 23:17
• You say $k$ is merely a field of characteristic zero, hence it could be the rationals or the p-adic numbers, neither of which will give rise to Cstar algebras. Do you actually know the definitions involved? In fact $A_1(k)$ cannot be a normed algebra, let alone a Cstar algebra. I fear that this contributes to an impression that you are trying to attack the Jacobian conjecture by finding things from various parts of maths to throw at the problem – Yemon Choi Sep 15 at 1:42