Puiseux Series – Example

We begin with a definition and an example before we state an important theorem.

Definition.  Puiseux series is a generalized power series in one variable, allowing fractional and negative exponents, of the form

 x(t) = \sum^{\infty}_{i=i_0} c_it^{\frac{1}{m}}, ~~i_0, m \in \mathbb{Z}, m>0, c_i \in \mathbb{C}.

We can view a Puiseux series x(t) in t as a Laurent series in t^{\frac{1}{m}}.  The set of Puiseux series over a field k forms a field denoted k\{\{t\}\}.  In fact, there is an even stronger result to be discussed shortly. The field of Puiseux series is a field with valuation.  A valuation on k\{\{t\}\} is a function val: k\{\{t\}\} \rightarrow \mathbb{R}\cup\infty with the properties [1]:

  1. val(a)=\infty if and only if a = 0,
  2. val(ab)=val(a)+val(b) and
  3. val(a+b)\geq \min\{val(a),val(b)\} for all a,b \in k\{\{t\}\}^*.

In other words, the valuation of an element in k\{\{t\}\} is the lowest exponent of a term with non vanishing coefficient in k. For instance, val(2-t^2+t^3)=0, val(t^7-12t^2)=2, and val(0)=\infty.

Example.  Consider the polynomial ring k\{\{t\}\}[x] and the polynomial f(t,x)=x^2+x-t.  We would like to find a Puiseux series solution x(t)\in k\{\{t\}\}.

The way we approximate the solution in terms of Puiseux series is analogous to Newton’s method for approximating a root of a univariate function.  Here however, instead of using the formal derivative of the function we use the Newton polygon, and we apply an analogous recursive method.  For further detail and examples on Newton’s method for an implicitly defined function y in terms of an indeterminate x see [3].

Similarly to Newton’s method we are only concerned with the lowest order terms.  We are looking for a solution of f(t,x) of the form

x(t)=b_0t^{\beta_0} + higher ~ order ~ terms.

Substituting this solution we obtain

f(t,x(t)) = -t + b_0t^{\beta_0}+b_0^2t^{2\beta_0}+\cdots,

where we do not take into account any of the higher order terms.  Of the lowest order terms we must have that a minimum valuation is attained at least twice.  That is \min\{1, \beta_0,2\beta_0\} occurs at least twice.  This gives rise to a system of linear inequalities

1=\beta_0\leq 2\beta_o, ~~~ 1=2\beta_0\leq\beta_0, ~~~ \beta_0=2\beta_0\leq 1.

Note that there are only two possible solutions for \beta_0, that is \beta_0=0 or $\latex \beta_0=1$. Since we are solving a quadratic polynomial in x over an algebraically closed field we expect to find two roots. The values of \beta_0 give us the lowest terms in each of the two roots of f(t,x). When \beta_0=0, the terms with lowest valuation are x^2+x=0, which results in x_1(t)=-1, and for \beta_0=1 the terms with lowest valuation are x-t=0, implying x_2(t)=t.
This same process can be done in terms of the Newton polygon. We can plot the points (i, val(a_i(t))), where a_i(t) are the coefficients of x^i and construct their convex hull in the positive quadrant. Then we select the edge with the slope with lowest negative slope, and the two endpoints of that edge are the terms with the lowest valuation.

Now, observe that x_1(t), x(2) are not solutions of the original equation, so we must continue this process for each solution. Let us take x_1(t)=-1+p where p is the adjestment to our approximation. Substitute x_1(t)=-1+p \text{~in~} f(t,x) and repeat the process above for p(t). Continue this process until it terminates or to reach desired number of interations. For this particular example, we obtain the following two Puiseux series solutions

x_1(t)=-1-t+t^2-2t^3+\cdots, ~~~ x_2(t)=t+t^2+2t^3+5t^4+\cdots.

Another example is f(t,x)=x^2-t^3+2t^2-t. The Puiseux series solution here only has two terms and they both have fractional exponenets, i.e. x_{1} = t^{1/2}-t^{3/2}, \text{~and~} x_2(t)=-t^{1/2}+t^{3/2}.

A natural question arises:  Can we always find a solution x(t) to a polynomial equation f(t,x) in the field of convergent Puiseux series?  The answer is the Newton-Puiseux Theorem (or just Puiseux Theorem).

References:

[1]  Diane Maclagan & Bernd Sturmfels, Introduction to Tropical Geometry

[2]  Bernd Sturmfels, Solving Systems of Polynomial Equations

[3]  Chris Christensen, Newton’s Method for Resolving Affected Equations

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