An irreducible polynomial F of degree  in the variable y with coefficients in a field  gives an algebraic extension L(x)[y]/(F) of , that is, it gives an algebraic function  which is a multivalued function. The algebraic function has  Puiseux expansions at x=a. Each Puiseux expansion is of the form  for some integer  ( is called the ramification index of the Puiseux expansion), and some integer  and elements  in the algebraic closure of .

The algcurves[algfun_series_sol] command determines those Puiseux expansions whose coefficients  have special properties ("nice" expansions). It can determine polynomial, rational, hypergeometric, d'Alembertian, m-sparse, or m-sparse m-hypergeometric sequences  (where m is an integer greater than  and can be given as an optional argument or, otherwise, be chosen as the maximal value for each expansion by the command).

If the point of expansion is given and is of the form x=a, the command returns a set of "nice" expansions at a. Otherwise, the command finds all the "nice" points. These are the points where the "nice" expansions exist. In this case, the command returns a set of lists ['Point'=the point , a set of "nice" expansions at ]. The point  is an algebraic number or the symbol _a  which denotes any regular point of the given algebraic function.

The command determines the field  from the input. The ground field  of the computation is the smallest field such that F and the given or constructed points of expansion are in L(x)[y]. The field  must be a rational number field or its algebraic extension by an algebraic number or several parameters.

The command builds a linear ordinary differential equation which the given algebraic function satisfies. Then it uses the package Slode for finding "nice" points and for building "nice" expansions for the constructed linear ordinary differential equation. The "nice" Puiseux expansion at the point x=a is represented in one of the following forms:

For a polynomial, rational, hypergeometric, or d'Alembertian sequence :

where  and  is a polynomial, rational, hypergeometric, or d'Alembertian term in .

For an m-sparse m-hypergeometric sequence :

where  and  is a hypergeometric term in .

For an m-sparse sequence :

where  is a homogeneous linear recurrence with polynomial coefficients for the sequence .

Each optional argument is of the type 'option' = 'value'. The following options are supported.

coeffs

Specifies the type of the coefficients; the default value is hypergeom.

dAlembertian, that is,  is annihilated by a linear recurrence operator that can be written as a composition of first-order operators (see LinearOperators).

hypergeom, that is,  is a rational function in  for all large enough .

rational, that is,  is a rational function in  for all large enough .

polynomial, that is,  is a polynomial in  for all large enough .

sparse, that is, for all large enough  only each m-th coefficient  is nonzero, where  is an integer (it is called the sparse order) and  is an integer such that .

mhypergeom, that is,  satisfies the property of the option sparse and  is a rational function in  for all large enough .

msparse_order

Specifies the sparse order. This option is given as 'msparse_order'=n, where n is an integer greater than 1. This option is useful only if one of the options 'coeffs'='mhypergeom' or 'coeffs'='sparse' is given.

mpoints_search

Specifies the algorithm for finding the "nice" points. This option is useful only in the case when the point of expansion is not specified and one of the options 'coeffs'='mhypergeom' or 'coeffs'='sparse' is given. The default value is full.

full, that is, the algorithm, which gives all "nice" points

quick, that is, the algorithm, which may miss some "nice" points but takes considerably less time than the full algorithm

RNF

Specifies the representation of . This option makes sense only if one of the options 'coeffs'='hypergeom' or 'coeffs'='mhypergeom' is given. The default value is false.

false, that is,  is represented as in the package LREtools.

true, that is,  is the first minimal representation of a hypergeometric term (see RationalNormalForms).