# What was Newton's road to his discovery of "Puiseux series" and "Newton polygon"?

In my opinion, one of Isaac Newton's greatest achievements in the "purer" aspects of mathematics was his discovery of Puiseux series; power series with fractional exponents. According to p.6 of the book "A singular mathematical promenade":

Strictly speaking, Newton did not provide proofs, but he did understand that locally an analytic curve consists of a finite number of branches, which are “graphs” of formal power series with rational exponents.

In his published work, Newton shows how the general method of power series with fractional exponents enables to present algebraic planar curves, which are more often given in implicit form $$F(x,y) = 0$$, in an explicit form $$y = f(x)$$. In a letter to Oldenburg he also describes the use of "Newton polygon" as a graphical aid in the calculation of Puiseux series. These spectacular achievements were made about 150 years before the developements by Laurent and Puiseux, and i think the generallity and usefulness of these methods somehow hides Newton's remarkable thought process behind what is apparently a practical motivation.

I have to say that the same book i quoted also says that Newton's results were the first important results toward understanding of the topology of an algebraic (or analytic) curve in the neighbourhood of a singular point. I have to admit that i don't really understand this matter (i.e development in Puiseux series), so the sentence i quoted might already give a kind of answer to the title question, but i need to see and understand it in my eyes in order to see the answer (i hope the answer to this question will give me some idea about those matters). Also, i'm not familiar with the orginization and chronology of Newton's works, so i find it hard to search for the interconnections in Newton's mathematical works.

Therefore, my questions are:

• Were Newton's discoveries related to his work on the classification of cubic curves? i suggest it because some cubic curves have "cusp singularities" so this might be a possible source of inspiration for Newton's discoveries.
• If the answer to the previous question is no, then what was Newton's road to these discoveries?
• What are the modern uses and connections of Puiseux series in mathematics?
• Newton very clearly described this in his papers, with many examples, and So texplained his intuition about this. The papers are available online in Engish. So nothing is "hidden" here. Jul 29 '20 at 18:04
• @AlexandreEremenko - O. k i'll delete the word "hidden" from the title of my question. Jul 29 '20 at 19:02
• As far as i understand the aim of HSM stack exchange, it should help exchange historical information and be a shortcut platform for answering such questions, although one can find the answers by himself (by investigating an historical riddle for days). I know that some of my questions seem rhetorical, but this is a direct result of my lack of knowledge. This site shouldn't help those who already know... Jul 29 '20 at 19:19
• There are some good pointers and references in the MO thread Newton and Newton polygon. Jul 30 '20 at 6:37