The newton_polygon function constructs the Newton polygon of a linear differential operator at the point $\mathrm{x0}$. The linear differential operator L corresponds to the differential equation $L\left(y\right)=0$. The equation $L\left(y\right)=0$ must be homogeneous and linear in y and its derivatives, and its coefficients must be rational functions in the variable $x$. The variable u must be a name and specifies the variable for the Newton polynomials. The variable x0 must be a rational or an algebraic number or the symbol infinity. If x0 is not passed as argument, then $\mathrm{x0}=0$ is assumed.

The Newton polygon of a linear differential operator at a point x0 is defined in the following way: for a rational function $f$, denote $\mathrm{v\_x0}\left(f\right)$ the valuation at the point x0, that is, the lowest power in the series expansion of $f$ at $\mathrm{x0}$: $f=f_0{\left(x-\mathrm{x0}\right)}^{\mathrm{v\_x0}\left(f\right)}+\mathrm{terms\; of\; higher\; order}$.  $f_0$ is called the leading coefficient of $f$ at $\mathrm{x0}$. For real numbers u,v let $Q\left(u\,v\right)$ be the subset of $R\mathrm{x}R$ defined by $\{\left(x,y\right)\|x\le u\mathrm{and}y\ge v\}$. Now consider a monomial $a_i\frac{d^i}{{\mathrm{dx}}^i}y\left(x\right)$ of $L$. Define $M\left(L\right)$ as the union of $Q\left(i\,\mathrm{v\_x0}\left(a_i\right)-i\right)$ for $i=0..\mathrm{order}\left(L\right)$. The Newton polygon is then the lower convex hull of the set $M\left(L\right)$. The slopes of the Newton polygon are non-negative rational numbers, and with each slope one associates the Newton polynomial $P_k\left(u\right)$. The length $l\left(k\right)$ of a slope $k$ is defined as the length of the projection of this slope onto the x-axis. $P_k$ is a polynomial in the new indeterminate u of degree $\frac{l\left(k\right)}{\mathrm{denom}\left(k\right)}$. Its monomials can be computed from the points which lie exactly on the slope $k$ and the leading coefficients of the corresponding $a_i\'s$. This Newton polynomial is called "reduced characteristic polynomial" in Barkatou, "Rational Newton Algorithm for computing formal solutions of linear differential equations", ISSAC'88.

The Newton polygon is an important tool for the classification of singular points of linear differential equations, because the slopes give information on the asymptotic behavior of the solutions near the singularity. It is also used in the computation of formal solutions; see also DEtools[formal_sols] and DEtools[regular_parts].

The output is a list of entries of the form $\left[k\,P_k\left(u\right)\right]$ where $k$ is a slope of the Newton polygon and $P_k$ is the associated Newton polynomial. Note that $P_0\left(u\right)$ is also called an indicial equation. The roots of $P_0\left(u\right)$ are called the exponents.

This function is part of the DEtools package, and so it can be used in the form newton_polygon(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[newton_polygon](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{ode}≔x^7\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\,x\right)-\left(x+x^7\right)\mathrm{diff}\left(y\left(x\right)\,x\right)-x^{9}y\left(x\right)$

$\mathrm{ode}≔x^7\left(\frac{{\ⅆ}^4}{\ⅆx^4}y\left(x\right)\right)-\left(x^7+x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-x^{9}y\left(x\right)$

$\mathrm{newton\_polygon}\left(\mathrm{ode}\,y\left(x\right)\,u\right)$

$\left[\left[0\,-u\right]\,\left[1\,u^3-1\right]\right]$

$\mathrm{newton\_polygon}\left(\mathrm{ode}\,y\left(x\right)\,u\,x=1\right)$

$\left[\left[0\,u^4-6u^3+11u^2-6u\right]\right]$

$\mathrm{newton\_polygon}\left(\mathrm{ode}\,y\left(x\right)\,u\,x=\mathrm{\infty }\right)$

$\left[\left[\frac{3}{2}\,u^2-1\right]\right]$

$\mathrm{ode}≔{\left(x^2+1\right)}^{5}x\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-{\left(x^2+1\right)}^3\mathrm{diff}\left(y\left(x\right)\,x\right)-x^{9}y\left(x\right)$

$\mathrm{ode}≔{\left(x^2+1\right)}^{5}x\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-{\left(x^2+1\right)}^3\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-x^{9}y\left(x\right)$

$\mathrm{newton\_polygon}\left(\mathrm{ode}\,y\left(x\right)\,u\,x=\mathrm{RootOf}\left({\mathrm{xx}}^2+1\,\mathrm{xx}\right)\right)$

$\left[\left[\frac{3}{2}\,32u+\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right]\right]$