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27.1 Definitions for Affine |

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__Function:__**fast_linsolve***([*`expr_1`, ...,`expr_m`], [`x_1`, ...,`x_n`])Solves the simultaneous linear equations

`expr_1`, ...,`expr_m`for the variables`x_1`, ...,`x_n`. Each`expr_i`may be an equation or a general expression; if given as a general expression, it is treated as an equation of the form

.`expr_i`= 0The return value is a list of equations of the form

`[`

where`x_1`=`a_1`, ...,`x_n`=`a_n`]`a_1`, ...,`a_n`are all free of`x_1`, ...,`x_n`.`fast_linsolve`

is faster than`linsolve`

for system of equations which are sparse.

__Function:__**grobner_basis***([*`expr_1`, ...,`expr_m`])Returns a Groebner basis for the equations

`expr_1`, ...,`expr_m`. The function`polysimp`

can then be used to simplify other functions relative to the equations.grobner_basis ([3*x^2+1, y*x])$ polysimp (y^2*x + x^3*9 + 2) ==> -3*x + 2

`polysimp(f)`

yields 0 if and only if`f`is in the ideal generated by`expr_1`, ...,`expr_m`, that is, if and only if`f`is a polynomial combination of the elements of`expr_1`, ...,`expr_m`.

__Function:__**set_up_dot_simplifications***(*`eqns`,`check_through_degree`)__Function:__**set_up_dot_simplifications***(*`eqns`)The

`eqns`are polynomial equations in non commutative variables. The value of`current_variables`

is the list of variables used for computing degrees. The equations must be homogeneous, in order for the procedure to terminate.If you have checked overlapping simplifications in

`dot_simplifications`

above the degree of`f`, then the following is true:`dotsimp (`

yields 0 if and only if`f`)`f`is in the ideal generated by the equations, i.e., if and only if`f`is a polynomial combination of the elements of the equations.The degree is that returned by

`nc_degree`

. This in turn is influenced by the weights of individual variables.

__Function:__**declare_weight***(*`x_1`,`w_1`, ...,`x_n`,`w_n`)Assigns weights

`w_1`, ...,`w_n`to`x_1`, ...,`x_n`, respectively. These are the weights used in computing`nc_degree`

.

__Function:__**nc_degree***(*`p`)Returns the degree of a noncommutative polynomial

`p`. See`declare_weights`

.

__Function:__**dotsimp***(*`f`)Returns 0 if and only if

`f`is in the ideal generated by the equations, i.e., if and only if`f`is a polynomial combination of the elements of the equations.

__Function:__**fast_central_elements***([*`x_1`, ...,`x_n`],`n`)If

`set_up_dot_simplifications`

has been previously done, finds the central polynomials in the variables`x_1`, ...,`x_n`in the given degree,`n`.For example:

set_up_dot_simplifications ([y.x + x.y], 3); fast_central_elements ([x, y], 2); [y.y, x.x];

__Function:__**check_overlaps***(*`n`,`add_to_simps`)Checks the overlaps thru degree

`n`, making sure that you have sufficient simplification rules in each degree, for`dotsimp`

to work correctly. This process can be speeded up if you know before hand what the dimension of the space of monomials is. If it is of finite global dimension, then`hilbert`

should be used. If you don't know the monomial dimensions, do not specify a`rank_function`

. An optional third argument`reset`

,`false`

says don't bother to query about resetting things.

__Function:__**mono***([*`x_1`, ...,`x_n`],`n`)Returns the list of independent monomials relative to the current dot simplifications of degree

`n`in the variables`x_1`, ...,`x_n`.

__Function:__**monomial_dimensions***(*`n`)Compute the Hilbert series through degree

`n`for the current algebra.

__Function:__**extract_linear_equations***([*`p_1`, ...,`p_n`], [`m_1`, ...,`m_n`])Makes a list of the coefficients of the noncommutative polynomials

`p_1`, ...,`p_n`of the noncommutative monomials`m_1`, ...,`m_n`. The coefficients should be scalars. Use`list_nc_monomials`

to build the list of monomials.

__Function:__**list_nc_monomials***([*`p_1`, ...,`p_n`])__Function:__**list_nc_monomials***(*`p`)Returns a list of the non commutative monomials occurring in a polynomial

`p`or a list of polynomials`p_1`, ...,`p_n`.

__Option variable:__**all_dotsimp_denoms**Default value:

`false`

When

`all_dotsimp_denoms`

is a list, the denominators encountered by`dotsimp`

are appended to the list.`all_dotsimp_denoms`

may be initialized to an empty list`[]`

before calling`dotsimp`

.By default, denominators are not collected by

`dotsimp`

.

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