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''Newton polygon''

A convex polygonal line, introduced by I. Newton in 1669 (see [[#References|[1]]]) to determine the exponents of the principal terms of algebraic functions. The process of finding successively the terms of the expansion of an algebraic function with the help of the Newton diagram is called the method of the Newton diagram. In more detail it was worked out by V. Puiseux [[#References|[2]]] and it is sometimes called the Puiseux diagram in mathematical literature. Before Puiseux, an algebraic version of the Newton diagram was studied by J.L. Lagrange [[#References|[3]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n0665201.png" /> be a pseudo-polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n0665202.png" />, that is, let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n0665203.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n0665204.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n0665205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n0665206.png" /> are complex variables, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n0665207.png" /> are complex numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n0665208.png" /> is a natural number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n0665209.png" /> are non-negative rational numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652011.png" />. As a rule one assumes that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652013.png" />, hence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652015.png" />. A solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652016.png" /> of the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

is sought for in the form of a series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652019.png" /> or, briefly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652021.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652022.png" />. To determine the possible values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652024.png" /> one substitutes (2) in (1), collects terms with equal powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652025.png" />, and equates to zero the coefficients of these powers.

The process begins with the term of lowest degree. As long as the exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652026.png" /> is not yet determined, there is no way of telling which of the resulting terms are lowest in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652027.png" />. However, the terms of lowest order are among the following:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652029.png" /> ranges over those values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652030.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652031.png" />. To annihilate the terms of lowest order one has to choose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652032.png" /> so that at least two of the exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652035.png" /> coincide and the remaining ones are to be not smaller. This argument leads to Newton's diagram.

In the plane one takes a rectangular Cartesian coordinate system and plots the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652039.png" /> ranges over the same values as in (3). Through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652040.png" /> one draws the line that coincides with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652041.png" />-axis and then rotates around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652042.png" /> anti-clockwise until it falls onto one of the plotted points, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652043.png" />. The tangent of the angle that the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652044.png" /> passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652046.png" /> makes with the negative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652047.png" />-axis is one of the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652048.png" />, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652049.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652050.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652051.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652052.png" /> is the point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652053.png" /> with largest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652054.png" />-coordinate and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652055.png" /> is rotated anti-clockwise around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652056.png" /> until it falls onto another one of the plotted points, say, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652057.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652058.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652059.png" /> be the line through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652061.png" />. The tangent of the angle between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652062.png" /> and the negative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652063.png" />-axis gives another possible value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652064.png" />. Continuing these constructions one obtains a convex polygonal line, which is called Newton's diagram.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n066520a.gif" />

Figure: n066520a

The value of the coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652065.png" /> is determined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652067.png" /> be the extreme points of a segment of the Newton diagram that determines one of the possible values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652068.png" />. To annihilate the terms of lowest order when (2) is substituted in (1) it is necessary and sufficient that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652069.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

where the prime in the sum denotes that summation is over those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652070.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652071.png" />. Equation (4) has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652072.png" /> non-zero roots (including multiplicity), that is, as many as the length of the projection of the relevant segment of the Newton diagram. Hence it is clear that by the method of the Newton diagram one obtains all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652073.png" /> values of the principal term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652074.png" /> in (2). By the same method one determines the next term in the expansion (2), etc. As a result, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066520/n06652075.png" /> solutions of (1) have the form (2), so-called Puiseux series (see [[Algebraic function|Algebraic function]]). The method of the Newton diagram is also applicable to the solution of differential equations.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Newton,   "The mathematical papers of I. Newton" , '''1–8''' , Cambridge Univ. Press  (1967–1981)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V. Puiseux,   "Récherches sur les fonctions algébriques"  ''J. Math. Pure Appl.'' , '''15'''  (1850)  pp. 365–480</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.L. Lagrange,   "Solution de quelques problèmes d'astronomie sphérique par le moyen des séries"  ''Nouv. Mém. Acad. Roy. Sci. Belles Lettres Berlin''  (1776)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.M. Vainberg,   V.A. Trenogin,   "Theory of branching of solutions of non-linear equations" , Noordhoff  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> , ''Isaac Newton 1643 - 1727; a collection of articles on the tercentenary of his birth'' , Moscow  (1943)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.D. [A.D. Bryuno] Bruno,   "Local methods in nonlinear differential equations" , Springer  (1989)  (Translated from Russian)</TD></TR></table>