determine Puiseux expansions of an algebraic function with "nice" coefficients - Maple Help

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algcurves[algfun_series_sol] - determine Puiseux expansions of an algebraic function with "nice" coefficients

Calling Sequence

algfun_series_sol(F, x, y, opts)

algfun_series_sol(F, x=a, y, opts)

Parameters

F

-

irreducible polynomial in two variables

x

-

name of the independent variable

y

-

name of the dependent variable

opts

-

sequence of optional arguments of the form 'option'='value', where option is one of coeffs, msparse_order, mpoints_search, or RNF

x=a

-

point of expansion

Description

• 

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

• 

The algcurves[algfun_series_sol] command determines those Puiseux expansions whose coefficients vn have special properties ("nice" expansions). It can determine polynomial, rational, hypergeometric, d'Alembertian, m-sparse, or m-sparse m-hypergeometric sequences vn (where m is an integer greater than 1 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, a set of "nice" expansions at x=A]. The point A is an algebraic number or the symbol _a  which denotes any regular point of the given algebraic function.

• 

The command determines the field L from the input. The ground field L 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 L 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:

1. 

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

vKxaKr+...+vμ1xaμ1r+n=μγnxanr

  

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

2. 

For an m-sparse m-hypergeometric sequence vn:

vKxaKr+...+vmμ+M1xamμ+M1r+n=μγnxamn+Mr

  

where 0<=M<m and γn is a hypergeometric term in n.

3. 

For an m-sparse sequence vn:

FPSstructvKxaKr&plus;...&plus;vmμ&plus;M1xamμ&plus;M1r&plus;n&equals;μvmn&plus;Mxamn&plus;Mr&comma;rec

  

where rec is a homogeneous linear recurrence with polynomial coefficients for the sequence vmn&plus;M.

Options

• 

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

– 

hypergeom, that is, vn&plus;1vn is a rational function in n for all large enough n.

– 

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

– 

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

– 

sparse, that is, for all large enough n only each m-th coefficient vmn&plus;M is nonzero, where 2m is an integer (it is called the sparse order) and M is an integer such that 0M<m.

– 

mhypergeom, that is, vn satisfies the property of the option sparse and vn&plus;1m&plus;Mvmn&plus;M is a rational function in n for all large enough n.

  

 

  

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 vn. 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, vn is represented as in the package LREtools.

– 

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

Examples

f:=x112x8&plus;x5y2x6&plus;2y2x3y2&plus;2yx32y1

f:=x112x8x6y2&plus;x5&plus;2x3y2&plus;2x3yy22y1

(1)

algcurves&lsqb;algfun_series_sol&rsqb;f&comma;x&equals;0&comma;y&comma;&apos;coeffs&apos;&equals;&apos;mhypergeom&apos;

1x5&sol;2x3x6x9_n&equals;4&infin;x6_n

(2)

f:=x5y61

f:=x5y61

(3)

algcurves&lsqb;algfun_series_sol&rsqb;f&comma;x&comma;y

Point&equals;0&comma;1x5&sol;6&comma;Point&equals;_a&comma;RootOf_Z6_a_n&equals;0&infin;1_a_n&Gamma;56&plus;_nx_a_n&Gamma;_n&plus;1_a&Gamma;56

(4)

f:=xy212yy2

f:=xy2y22y1

(5)

algcurves&lsqb;algfun_series_sol&rsqb;f&comma;x&comma;y&comma;&apos;coeffs&apos;&equals;&apos;polynomial&apos;

Point&equals;0&comma;_n&equals;0&infin;x_n

(6)

algcurves&lsqb;algfun_series_sol&rsqb;f&comma;x&equals;4&comma;y&comma;&apos;coeffs&apos;&equals;&apos;dAlembertian&apos;

112_n&equals;0&infin;121_n3_n&pi;&plus;1_n&plus;13_n&plus;1_n1&equals;0_n13_n1&Gamma;_n1&plus;124_n1&Gamma;_n1&plus;2x4_n&pi;&comma;112_n&equals;0&infin;1_n3_n&plus;1_n1&equals;0_n13_n1&Gamma;_n1&plus;124_n1&Gamma;_n1&plus;2&plus;41_n&plus;13_n&pi;x4_n&pi;

(7)

See Also

algcurves, algcurves[puiseux], algcurves[singularities], LinearOperators, LREtools, LREtools[hypergeomsols], RationalNormalForms, RationalNormalForms[MinimalRepresentation], Slode, Slode[candidate_mpoints], Slode[candidate_points]


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