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

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

We can view a Puiseux series in as a *Laurent series* in The set of Puiseux series over a field forms a field denoted . 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 is a function with the properties [1]:

- if and only if
- and
- for all

In other words, the valuation of an element in is the lowest exponent of a term with non vanishing coefficient in . For instance, , , and

**Example.** Consider the polynomial ring and the polynomial . We would like to find a Puiseux series solution .

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 in terms of an indeterminate see [3].

Similarly to Newton’s method we are only concerned with the lowest order terms. We are looking for a solution of of the form

Substituting this solution we obtain

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 occurs at least twice. This gives rise to a system of linear inequalities

Note that there are only two possible solutions for , that is or $\latex \beta_0=1$. Since we are solving a quadratic polynomial in over an algebraically closed field we expect to find two roots. The values of give us the lowest terms in each of the two roots of . When , the terms with lowest valuation are , which results in , and for the terms with lowest valuation are , implying .

This same process can be done in terms of the Newton polygon. We can plot the points , where are the coefficients of 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 are not solutions of the original equation, so we must continue this process for each solution. Let us take where is the adjestment to our approximation. Substitute and repeat the process above for . 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

Another example is . The Puiseux series solution here only has two terms and they both have fractional exponenets, i.e. .

A natural question arises: *Can we always find a solution to a polynomial equation 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