Is there a general efficient method/approach to solve a system of $N$ cubic equations in reals such that the number of variables $v$ are less than $N$.
I am aware of the simple approach for linear equations but unclear if something exists for cubic equation. Can someone please explain and provide a pointer?
Lets assume a simple case where there are at most $k$ terms in each equation.
Eg of a very trivial case:
$a^3+b=9$
$a^2+b=5$
$a+b=3$
here $N=3, v=2, k=2$.
I don't have many things to say, but it's too long for a comment so I write here.
To solve the rectangular system of nonlinear equations:
$$ \tag{A} \begin{cases} a^3 + b = 9 \\ a^2 + b = 5 \\ a + b = 3 \\ \end{cases} $$
I would determine candidate solutions by solving the squared system of nonlinear equations:
$$ \tag{B} \begin{cases} a^2 + b = 5 \\ a + b = 3 \\ \end{cases} $$
i.e. $(x,y) = (-1,4)$ or $(x,y) = (2,1)$. So, given that:
$$ (-1)^3 + 4 \ne 9, \quad \quad \quad \quad \quad 2^3 + 1 = 9 $$
I would conclude that the unique solution of $(A)$ is $(x,y)=(2,1)$.
I would have reached the same conclusion considering other pairs of equations, but probably having to carry out some more calculations. I don't know any algorithms for choosing the optimal squared systems.
On the other hand, what I know for sure is that the only systems of equations that we know how to solve efficiently are the linear ones (square or rectangular), while for the nonlinear ones (square or rectangular) things can be done much more complicated.
In general, we refer to squared systems of nonlinear equations and in this context we need to look the problem we want to solve in the face and on the basis of this we begin to dig into the mine of methods present in the literature. In particular, we can refer to two extreme cases.
The first refers to those systems in which we already have an idea of where the geometrically isolated solutions are located and therefore we can use the fastest method, i.e. the Newton-Raphson Method, since it requires a trigger point close enough to the desired one, otherwise it's probable that it will not converge or at least it will not do so immediately.
The second refers to those systems in which we have no idea where the geometrically isolated solutions are located and therefore we must equip ourselves with an "infallible convergent method"; an example could be the Homotopy Continuation Method. The price to pay, however, lies in the convergence time which will necessarily be longer.
On the other hand, then there are many other intermediate methods to choose from depending on the specific problem you want to solve. For example, the last method mentioned is excellent in the case of algebraic systems, while for non-algebraic ones it's necessary to fall back on metaheuristic methods such as the Differential Evolution Method.
In conclusion, what can I say, the matter is really very complicated, so try to focus your problem well and dig in this sense, don't expect to find the miraculous method immediately because in this way it's likely that you will lose from the start. Searching through the questions on my Mathematica profile you will find some toy codes of the above methods.