An irreducible $P$ with infinitely many real points generates the ideal of polynomials vanishing on them

Let $P\in\mathbb C[X,Y]$ be irreducible. Writing points of the real plane as $z\in\mathbb C$, the real locus of $P$ in the coordinates $X=z$, $Y=\bar z$ is $Z(P)=\{z\in\mathbb C:P(z,\bar z)=0\}$. Let $\Delta_P=\{(z,\bar z):z\in Z(P)\}\subseteq\mathbb C^2$ be its image under $z\mapsto(z,\bar z)$, and let $I(\Delta_P)=\{Q\in\mathbb C[X,Y]:Q(v)=0\ \text{for all}\ v\in\Delta_P\}$ be the ideal of the polynomials vanishing on $\Delta_P$.

If $Z(P)$ is infinite, then

the principal ideal of $\mathbb C[X,Y]$ generated by $P$.

This is the algebraic content of the Zariski density of an infinite real locus in the complex curve $P=0$, which the note invokes in the proofs of Lemma 3 and Theorem 2. For the polynomial $P_\alpha$ of equation (6), whose real locus is $C_{m,\alpha}$ and which satisfies the hypotheses by Lemma 4 when $|\alpha|=1$ and $\alpha\notin\mathbb R$, it says that the polynomials vanishing at all points $(z,\bar z)$ with $z\in C_{m,\alpha}$ are exactly the multiples of $P_\alpha$.

Formalization Note: $I(\cdot)$ is Mathlib's MvPolynomial.vanishingIdeal ℂ, and irreducibility is taken in MvPolynomial (Fin 2) ℂ.