I want to calculate the discriminant set of a function (germ) $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ based on the $\mathbb{R} \rightarrow \mathbb{R}$ example of this article. The function is defined as \begin{equation} f(x,y) = \begin{pmatrix} x^2 - y^2 \\ 2 x y \end{pmatrix} \end{equation} with $x,y \in \mathbb{R}$. This function defines a vector field around the origin of $\mathbb{R}^2$ and the vector field itself winds around the origin twice. One can define a perturbation of this function by introducing the following so-called unfolding $f_{\mathbf{t}}$ \begin{equation} f_{\mathbf{t}} = \begin{pmatrix} x^2 - y^2 \\ 2 x y \end{pmatrix} + \begin{pmatrix} t_1 x + t_2 y \\ t_3 x + t_4 y\end{pmatrix} + \begin{pmatrix} t_5 \\ t_6\end{pmatrix} \end{equation} where $t_i \in \mathbb{R}$ and they are assumed to be small $t_i \ll 1$. We can characterize the unfolding of $f$ by a vector $\mathbf{t} \in \mathbb{R}^6$, $\mathbf{t} = (t_1, t_2, t_3, t_4, t_5, t_6)$. For some points in this $\mathbb{R}^6$ the above unfolding have 2 points for which $f_{\mathbf{t}}(x,y) = 0$, while for some other points of $\mathbb{R}^6$ there are 4 such points where $f_{\mathbf{t}}$ vanishes.
What I am interested in is the boundary of these regions around the origin. This boundary is called the discriminant set (or set-germ in my case).
A similar scenario can be seen on Fig.2 of the cited article, where the function is a simple $\mathbb{R} \rightarrow \mathbb{R}$ mapping and the unfolding is defined by two parameters only : \begin{equation} f_\mathbf{t} = x^3 + p x + q \end{equation} with $\mathbf{t} = (p,q)$.
I know the definition of the discriminant set:
Def: The discriminant set of $f$ is the set of those points $\mathbf{t} \in \mathbb{R}^6$ (in our case) for which $\exists (x,y) \in \mathbb{R}^2 $ such that
$f_\mathbf{t}(x,y) = 0$ and the determinant of the Jacobian of the function $g(x,y,\mathbf{t}) = (f_\mathbf{t}(x,y), \mathbf{t}) $ vanishes.
In the above definition the function $g$ is defined to be a $\mathbb{R}^2 \times \mathbb{R}^6 \rightarrow \mathbb{R}^8$ mapping such that for each $(x,y) \in \mathbb{R}^2$ and $\mathbf{t} \in \mathbb{R}^6$ is mapped to $(f_{\mathbf{t},1}(x,y),f_{\mathbf{t},2}(x,y), t_1, t_2, t_3,t_4,t_5,t_6)$ and $f_{\mathbf{t},j}$ denotes the $j$-th component of the unfolding $f_\mathbf{t}$.
I have worked on the algebra and the constraint that characterize the discriminant set seem to be unsolvable.
Any suggestions on how to tackle such a problem without dealing with the equations themselves?
Much simplification occurs if we use complex notation from the outset. The unperturbed map $(x,y)\to (x^2-y^2, 2x y)$ can be written in complex notation as $z=x+iy\to w= u+iv=z^2$. The perturbation to this map is a general affine map from the $z$ plane to the $w$ plane, and this perturbing map can be written as $ z\to w=Az + B\overline z -C$
where $(A,B,C)$ are three arbitrary complex "control parameters" , which are the arbitrary constants in the unfolding space.These are equivalent to 6 real control parameters. (Note that the presence of $\overline z$ indicates that the typical linear perturbation is not holomorphic (aka not analytic.) (The analysis of complex-valued mappings that are built with such terms that are not analytic is an extension of classical 19th century complex analysis that has been developed extensively in the last century but is generally not discussed in the undergraduate curriculum. The book by Krantz cited below is a good introduction to the modern theory.) Thus the full expression for the fully perturbed map can be written in complex notation as
$z\to w=z^2 + A z+ B \overline z -C$.
My belief is that you want to understand under what circumstances the solutions of the perturbed equation $w(z)=0$ abruptly change character as we perturb the control parameters in unfolding space. This transition occurs when what were formerly two distinct roots appear to merge into a single solution of higher multiplicity (repeated root) If that interpretation is correct, then here is the next step. The equation $w=0$ means $ z^2 + Az + B\overline z=C$. Thus we can regard the relation $C= f(z)=z^2+ A z+ B\overline z$ as a map from the $z$ plane to the $C$ plane, and we want to see for which values of $A$ and $B$ there occurs a solution of multiplicity two or more in the $z$ plane. (One can also regard this as a situation in which the inverse function theorem breaks down: there fails to be a simple smoothly-varying local 1-1 way to assign exactly one single solution $z$ to a known value of $C$ because two solutions pinch together.) The inverse function theorem fails when the Jacobian of the map $z\to C$ is zero. In other words, the mapping $z\to C$ has a critical point.
So we need to find the critical points of $f$ in the $z$ plane, and then compute their image under the map $C=f(z)$. Henceforth we will occasionally write the map $C=f(z)$ instead as $w=f(z)$ since that is more traditional notation when analyzing complex-valued mappings.
The critical points. The Jacobian degenerates (i.e., the map $z\to w$ has a critical point as a function of $z$) when the differential $dz\to dw= (2 z+A) dz + B \ d \overline z$ has nontrivial kernel. Writing the nonzero element in the kernel as $dz= r e^{\theta}$ we see after some algebra (cancelling $r$) that this requires $ (2z+A) e^{i\theta} + B e^{-i\theta}=0$. Solutions exist iff $2z+A =-B e^{-2 i \theta}$ where $\theta$ is arbitrary. Thus we can parametrize the critical points in the $z$ plane nicely as $ z= \frac{-1}{2} [ A+ B e^{-2 i \theta}]$. As we vary the variable $\tau=-2\theta$ this parametrizes a circle in the $z$ plane centered at $z=-A/2$ of radius $|B|/2$. Denote this critical circle in the $z$ plane by $\Gamma =\Gamma (A,B)$. (Below we will assume it is parametrized as a function of $\tau=-2 \theta$.)
The critical values of $f$. Treating $C$ as a function of $z$ and the controls $A,B$, the critical values of $C$ are just the values obtained by evaluating the function $C= f( z: A,B)$ when $ z= \frac{-1}{2} [ A+ B e^{-2 i \theta}]$ lies on the critical circle $\Gamma$ determined by $A$ and $B$. Thus you are led to consider the subset of control space consisting of all points of the form $ (A,B,C)= (A,B, f(z, A,B)$ evaluated at all values of $z\in \Gamma(A,B)$, parametrized as a function of $\tau$.
One can see that as a function of $\tau=2\theta$, $C$ is a complex Fourier series. The various complex amplitudes are certain explicit linear and quadratic terms in $A$ and $B$. In fact
$$C=-(A^2/4) - (\overline A B)/2 - 1/2 B \overline B e^{-i \tau} + 1/4 B^2 e^{2 i \tau}$$
The Fourier series $\tau \to C(\tau)$ traces a deltoid curve that has three cusp points forming an equilateral triangle. The cusps occur where $dC/d\tau=0$. Below is a plot of a typical example.
P.S. The corrected picture below now should match the parameter constants, as discussed in the comments sections.
P.P.S Ahlfors has an introductory section (see p. 6 excerpted below) Lectures on Quasiconformal maps that has a nice geometric discussion of the significance of the terms in the complex expansions $dw= f_z dz+ f_{\overline z} \overline dz$ and he explicitly computes the Jacobian $J(f)$ in terms of these two parameters. (This notation for the terms in this expansion is sometimes attributed to Wirtinger.). He shows that the Jacobian vanishes precisely when $ |f_z|= | f_{\overline z}|$ which explains why the critical circle computed earlier is given by $|2z+A|= |B|$.