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<h1 class="chapter"> 27. Affine </h1>
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<tr><td align="left" valign="top"><a href="#SEC92">27.1 Definitions for Affine</a></td><td> </td><td align="left" valign="top">
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<h2 class="section"> 27.1 Definitions for Affine </h2>
<dl>
<dt><u>Function:</u> <b>fast_linsolve</b><i> ([<var>expr_1</var>, ..., <var>expr_m</var>], [<var>x_1</var>, ..., <var>x_n</var>])</i>
<a name="IDX836"></a>
</dt>
<dd><p>Solves the simultaneous linear equations <var>expr_1</var>, ..., <var>expr_m</var>
for the variables <var>x_1</var>, ..., <var>x_n</var>.
Each <var>expr_i</var> may be an equation or a general expression;
if given as a general expression, it is treated as an equation of the form <code><var>expr_i</var> = 0</code>.
</p>
<p>The return value is a list of equations of the form
<code>[<var>x_1</var> = <var>a_1</var>, ..., <var>x_n</var> = <var>a_n</var>]</code>
where <var>a_1</var>, ..., <var>a_n</var> are all free of <var>x_1</var>, ..., <var>x_n</var>.
</p>
<p><code>fast_linsolve</code> is faster than <code>linsolve</code> for system of equations which
are sparse.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>grobner_basis</b><i> ([<var>expr_1</var>, ..., <var>expr_m</var>])</i>
<a name="IDX837"></a>
</dt>
<dd><p>Returns a Groebner basis for the equations <var>expr_1</var>, ..., <var>expr_m</var>.
The function <code>polysimp</code> can then
be used to simplify other functions relative to the equations.
</p>
<table><tr><td> </td><td><pre class="example">grobner_basis ([3*x^2+1, y*x])$
polysimp (y^2*x + x^3*9 + 2) ==> -3*x + 2
</pre></td></tr></table>
<p><code>polysimp(f)</code> yields 0 if and only if <var>f</var> is in the ideal generated by
<var>expr_1</var>, ..., <var>expr_m</var>, that is,
if and only if <var>f</var> is a polynomial combination of the elements of
<var>expr_1</var>, ..., <var>expr_m</var>.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>set_up_dot_simplifications</b><i> (<var>eqns</var>, <var>check_through_degree</var>)</i>
<a name="IDX838"></a>
</dt>
<dt><u>Function:</u> <b>set_up_dot_simplifications</b><i> (<var>eqns</var>)</i>
<a name="IDX839"></a>
</dt>
<dd><p>The <var>eqns</var> are
polynomial equations in non commutative variables.
The value of <code>current_variables</code> is the
list of variables used for computing degrees. The equations must be
homogeneous, in order for the procedure to terminate.
</p>
<p>If you have checked overlapping simplifications in <code>dot_simplifications</code>
above the degree of <var>f</var>, then the following is true:
<code>dotsimp (<var>f</var>)</code> yields 0 if and only if <var>f</var> is in the
ideal generated by the equations, i.e.,
if and only if <var>f</var> is a polynomial combination
of the elements of the equations.
</p>
<p>The degree is that returned by <code>nc_degree</code>. This in turn is influenced by
the weights of individual variables.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>declare_weight</b><i> (<var>x_1</var>, <var>w_1</var>, ..., <var>x_n</var>, <var>w_n</var>)</i>
<a name="IDX840"></a>
</dt>
<dd><p>Assigns weights <var>w_1</var>, ..., <var>w_n</var> to <var>x_1</var>, ..., <var>x_n</var>, respectively.
These are the weights used in computing <code>nc_degree</code>.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>nc_degree</b><i> (<var>p</var>)</i>
<a name="IDX841"></a>
</dt>
<dd><p>Returns the degree of a noncommutative polynomial <var>p</var>. See <code>declare_weights</code>.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>dotsimp</b><i> (<var>f</var>)</i>
<a name="IDX842"></a>
</dt>
<dd><p>Returns 0 if and only if <var>f</var> is in the ideal generated by the equations, i.e.,
if and only if <var>f</var> is a polynomial combination of the elements of the equations.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>fast_central_elements</b><i> ([<var>x_1</var>, ..., <var>x_n</var>], <var>n</var>)</i>
<a name="IDX843"></a>
</dt>
<dd><p>If <code>set_up_dot_simplifications</code> has been previously done, finds the central polynomials
in the variables <var>x_1</var>, ..., <var>x_n</var> in the given degree, <var>n</var>.
</p>
<p>For example:
</p><table><tr><td> </td><td><pre class="example">set_up_dot_simplifications ([y.x + x.y], 3);
fast_central_elements ([x, y], 2);
[y.y, x.x];
</pre></td></tr></table>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>check_overlaps</b><i> (<var>n</var>, <var>add_to_simps</var>)</i>
<a name="IDX844"></a>
</dt>
<dd><p>Checks the overlaps thru degree <var>n</var>,
making sure that you have sufficient simplification rules in each
degree, for <code>dotsimp</code> 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 <code>hilbert</code> should be used. If you
don't know the monomial dimensions, do not specify a <code>rank_function</code>.
An optional third argument <code>reset</code>, <code>false</code> says don't bother to query
about resetting things.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>mono</b><i> ([<var>x_1</var>, ..., <var>x_n</var>], <var>n</var>)</i>
<a name="IDX845"></a>
</dt>
<dd><p>Returns the list of independent
monomials relative to the current dot simplifications of degree <var>n</var>
in the variables <var>x_1</var>, ..., <var>x_n</var>.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>monomial_dimensions</b><i> (<var>n</var>)</i>
<a name="IDX846"></a>
</dt>
<dd><p>Compute the Hilbert series through degree <var>n</var> for the current algebra.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>extract_linear_equations</b><i> ([<var>p_1</var>, ..., <var>p_n</var>], [<var>m_1</var>, ..., <var>m_n</var>])</i>
<a name="IDX847"></a>
</dt>
<dd><p>Makes a list of the coefficients of the noncommutative polynomials <var>p_1</var>, ..., <var>p_n</var>
of the noncommutative monomials <var>m_1</var>, ..., <var>m_n</var>.
The coefficients should be scalars. Use <code>list_nc_monomials</code> to build the list of
monomials.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>list_nc_monomials</b><i> ([<var>p_1</var>, ..., <var>p_n</var>])</i>
<a name="IDX848"></a>
</dt>
<dt><u>Function:</u> <b>list_nc_monomials</b><i> (<var>p</var>)</i>
<a name="IDX849"></a>
</dt>
<dd><p>Returns a list of the non commutative monomials occurring in a polynomial <var>p</var>
or a list of polynomials <var>p_1</var>, ..., <var>p_n</var>.
</p>
</dd></dl>
<dl>
<dt><u>Option variable:</u> <b>all_dotsimp_denoms</b>
<a name="IDX850"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>all_dotsimp_denoms</code> is a list,
the denominators encountered by <code>dotsimp</code> are appended to the list.
<code>all_dotsimp_denoms</code> may be initialized to an empty list <code>[]</code>
before calling <code>dotsimp</code>.
</p>
<p>By default, denominators are not collected by <code>dotsimp</code>.
</p>
</dd></dl>
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