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Problem Statement

My question is related to this one: Existence and uniqueness of solutions to a system of non-linear equations.

However it is more specific and I think it is a relatively easy case so maybe more light can be shed on it.

I have a set of equations which has a following form:

$$ k^* \left(\alpha k \alpha ^*+\beta l \alpha ^*\right)+l^* \left(\alpha k \beta ^*+\beta l \beta ^*\right)=\epsilon \epsilon ^*\\ p^* \left(\alpha k \alpha ^*+\beta l \alpha ^*\right)+q^* \left(\alpha k \beta ^*+\beta l \beta ^*\right)=\epsilon \zeta ^*\\ p^* \left(\alpha p \alpha ^*+\beta q \alpha ^*\right)+q^* \left(\alpha p \beta ^*+\beta q \beta ^*\right)=\zeta \zeta ^* $$

I consider latin letters as unknowns, while greek letters as complex numbers which are provided beforehand. The star stand for the conjugate, and my equations are over the complex field.

The case presented is therefore has three equations and four unknowns thus there's no reason to rule out the existence of a solution or even multiple solutions. Generally I may have more equations and more variables, but the general form of equations which are quadratic in variables will remain.

My questions are:

  1. Can the equations be solved? Since I don't have any periodic functions I guess that if multiple solutions exist there are only a finite number of them. Is there a way to understand how many solutions are there?
  2. Is there way to do some analysis symbolically without providing numeric values for the parameters and resulting to numeric techniques?
  3. I tried so treat to problem as an optimization problem (details below), and used Mathematica try and solve it. I'm not sure to do all gradient based algorithms extend to the complex domain, and apparently nor does Mathematica. Can they be extended? Can I face the issue by replacing each each complex variable by two variables corresponding to its real and imaginary part?

Any solutions using concepts related optimization or other approaches are welcome. I'll also appreciate references in the right direction.

My attempt so far:

I saw in some answer on this site (sadly I can't find it now for refernce) a suggestion to treat the problem as an optimization problem. Each equation can be written as a homogeneous equation by moving the LHS to the RHS, and then each equation could be thought of a a function of our variables. Thus by defining $x=(k,l,p,q)$ we get:

$$ F_1(x)=k^* \left(\alpha k \alpha ^*+\beta l \alpha ^*\right)+l^* \left(\alpha k \beta ^*+\beta l \beta ^*\right)-\epsilon \epsilon ^*\\ F_2(x)=p^* \left(\alpha k \alpha ^*+\beta l \alpha ^*\right)+q^* \left(\alpha k \beta ^*+\beta l \beta ^*\right)-\epsilon \zeta ^*\\ F_3(x)=p^* \left(\alpha p \alpha ^*+\beta q \alpha ^*\right)+q^* \left(\alpha p \beta ^*+\beta q \beta ^*\right)-\zeta \zeta ^* $$

As seen my functions are generally complex valuers functions and I wish to minimize the magnitude of each of them. Thus I defined a cost function:

$$ \phi(x)=F_1(x)F_1^*(x)+F_2(x)F_2^*(x)+F_3(x)F_3^*(x) $$

My cost function is therefore a real valued positive semidefinite function of the complex valued argument $x$, and I wish to minimize it. When I try to use Mathematica built-in functions to minimize my cost function FindMinimum is able to find a numeric solution upon using numeric values for parameters. However, regardless of chosen parameters (which I tried) the returned set consists of real numbers, which suggests to me that maybe it doesn't look for complex ones. Other functions return an error which states that only real valued functions and variables are allowed.

Bernard
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Yair M
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1 Answers1

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I wrote out your equations in terms of real and imaginary parts ($p = p_1 + i p_2$ etc.), and using Maple found a Groebner basis.

It seems this system has Hilbert dimension $5$, a maximal independent set being $l_1, l_2, p_2, q_1, q_2$. This does not take into account the requirement of the real and imaginary parts being real. However, it seems that in general you have a solution space of real dimension $5$, at least in some regions of parameter space. For example, taking $\alpha = 1+2i$, $\beta=2+3i$, $\epsilon = 3+4i$, $\zeta=4+5i$, we get a solution

$$ \eqalign{{k_1}&=-\frac85 l_1 -\frac15 l_2 + \frac{32}{41} p_1 - \frac{1}{41} p_2 + \frac{257}{205} q_1 + \frac{24}{205} q_2\cr k_2 &= \frac{1}{5} l_1 - \frac{8}{5} l_2 + \frac{1}{41} p_1 + \frac{32}{41} p_2 - \frac{24}{205} q_1 + \frac{257}{205} q_2\cr p_1 &= - \frac{8}{5} q_1 - \frac{1}{5} q_2 + \frac{1}{5} \sqrt{205 - 25 p_2^2 + 10 p_2 q_1 - 80 p_2 q_2 - q_1^2 + 16 q_1 q_2 - 64 q_2^2} } $$

Note that the quantity inside the square root is positive for real $( p_2, q_1, q_2)$ in a neighbourhood of $(0,0,0)$.

Robert Israel
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