$\operatorname{LMOpConvexHull}(L, x, M, b)$
libname := libname, FileTools:-JoinPath(["maple","lib","dcfun.mla"],base=homedir):
with(dcfun):
We consider the following linear Mahler operator $L$.
b := 2;
$$2$$
L := x^3*((x + 2*x^2) + ( 1 - x)*M + (10 + x^3)*M^2 + x^4*M^3 + x^12*M^4);
$$x^{3} \left(x +2 x^{2}+\left(1-x \right) M +\left(x^{3}+10\right) M^{2}+x^{4} M^{3}+x^{12} M^{4}\right)$$
We draw its Newton diagram and its lower and upper Newton polygons.
LMOpNewtonPolygon(L, 0, x, M, b);
$$$$
We now calculate its upper and lower Newton polygons. (Obviously, the previous drawing used this calculation.)
LMOpConvexHull(L, x, M, b, 'output' = 'geometry');
$$[[1, 4], [2, 3], [4, 3], [8, 7], [16, 15]], [[1, 5], [16, 15]]$$
The lower Newton polygon of $L$ has three edges. We compute the associated sequence of characteristic polynomials. The output of the first command is a list of three records.
ch := LMOpConvexHull(L, x, M, b, 'direction' = 'lower', 'output' = 'charpolys', 'variable' = 'alpha');
$$[\textbf{module}() \,\textbf{export} \,abscissa,\,ordinate,\,slope,\,intercept,\,charpoly; \, \textbf{end}, \textbf{module}() \,\textbf{export} \,abscissa,\,ordinate,\,slope,\,intercept,\,charpoly; \, \textbf{end}, \textbf{module}() \,\textbf{export} \,abscissa,\,ordinate,\,slope,\,intercept,\,charpoly; \, \textbf{end}, \textbf{module}() \,\textbf{export} \,abscissa,\,ordinate,\,slope,\,intercept,\,charpoly; \, \textbf{end}]$$
seq(r:-charpoly, r = ch);
$$1+\alpha, 1+10 \alpha, \alpha^{2}+\alpha +10, 0$$