How many equations for a system of non-linear equations?

Question

Let's suppose we have a system of n non-linear equations with m variables. How many equations does my system need to produce a single solution? Is there a theorem that states this?

Answer 1

Normally you need as many equations as variables. The fact that the equations are not linear is not important because you can linearize them in the vicinity of a root. However, you can have cases like $(x-1)^2+(y-2)^2=0$, where one equation is enough for two variables in the reals. It is not enough in the complex field.

Answer 2

Nonlinear functions are so extremely varied, that there is no general rule.

For instance, with just one equation in one variable in the reals,

$$\sin x-x=0$$ has a single solution.

$$2\sin x-x=0$$ has a three solutions.

$$\sin x-1=0$$ has an infinity of discrete solutions.

$$\sin x-2=0$$ has no solution.

$$\sin x-\sin |x|=0$$ has a whole half-line of solutions plus isolated ones.

$$\sin x-|\sin x|=0$$ has an infinity of intervals of solutions.

For

$$\Re\left(\zeta\left(a+ix\right)\right)=0$$ we are unsure about the existence of solutions when $a\ne\frac12$.

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As you can imagine, situations can be much more complex in higher dimensions.

It took three centuries to settle the case of

$$\lfloor x\rfloor^{\lfloor n\rfloor}+\lfloor y\rfloor^{\lfloor n\rfloor}=\lfloor z\rfloor^{\lfloor n\rfloor}.$$