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<h1 class="chapter"> 24. Affine </h1>

<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC121">24.1 Introduction to Affine</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC122">24.2 Functions and Variables for Affine</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
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<p><a name="Item_003a-Introduction-to-Affine"></a>
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<h2 class="section"> 24.1 Introduction to Affine </h2>

<p><code>affine</code> is a package to work with groups of polynomials.
</p>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Polynomials">Polynomials</a> &middot;
<a href="maxima_105.html#Category_003a-Groebner-bases">Groebner bases</a> &middot;
<a href="maxima_105.html#Category_003a-Share-packages">Share packages</a> &middot;
<a href="maxima_105.html#Category_003a-Package-affine">Package affine</a>
</div>
</p>

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<h2 class="section"> 24.2 Functions and Variables for Affine </h2>

<p><a name="fast_005flinsolve"></a>
<a name="Item_003a-fast_005flinsolve"></a>
</p><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="IDX919"></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>
<p><code>load(affine)</code> loads this function.
</p>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Package-affine">Package affine</a>
</div>
</p></dd></dl>

<p><a name="grobner_005fbasis"></a>
<a name="Item_003a-grobner_005fbasis"></a>
</p><dl>
<dt><u>Function:</u> <b>grobner_basis</b><i> ([<var>expr_1</var>, ..., <var>expr_m</var>])</i>
<a name="IDX920"></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>
<pre class="example">grobner_basis ([3*x^2+1, y*x])$

polysimp (y^2*x + x^3*9 + 2) ==&gt; -3*x + 2
</pre>
<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>
<p><code>load(affine)</code> loads this function.
</p>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Package-affine">Package affine</a>
</div>
</p></dd></dl>

<p><a name="set_005fup_005fdot_005fsimplifications"></a>
<a name="Item_003a-set_005fup_005fdot_005fsimplifications"></a>
</p><dl>
<dt><u>Function:</u> <b>set_up_dot_simplifications</b><i> &nbsp;<br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>set_up_dot_simplifications</tt> (<var>eqns</var>, <var>check_through_degree</var>) &nbsp;<br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>set_up_dot_simplifications</tt> (<var>eqns</var>)</i>
<a name="IDX921"></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>
<p><code>load(affine)</code> loads this function.
</p>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Package-affine">Package affine</a>
</div>
</p></dd></dl>

<p><a name="declare_005fweights"></a>
<a name="Item_003a-declare_005fweights"></a>
</p><dl>
<dt><u>Function:</u> <b>declare_weights</b><i> (<var>x_1</var>, <var>w_1</var>, ..., <var>x_n</var>, <var>w_n</var>)</i>
<a name="IDX922"></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>
<p><code>load(affine)</code> loads this function.
</p>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Package-affine">Package affine</a>
</div>
</p></dd></dl>

<p><a name="nc_005fdegree"></a>
<a name="Item_003a-nc_005fdegree"></a>
</p><dl>
<dt><u>Function:</u> <b>nc_degree</b><i> (<var>p</var>)</i>
<a name="IDX923"></a>
</dt>
<dd><p>Returns the degree of a noncommutative polynomial <var>p</var>.  See <code>declare_weights</code>.
</p>
<p><code>load(affine)</code> loads this function.
</p>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Package-affine">Package affine</a>
</div>
</p></dd></dl>

<p><a name="dotsimp"></a>
<a name="Item_003a-dotsimp"></a>
</p><dl>
<dt><u>Function:</u> <b>dotsimp</b><i> (<var>f</var>)</i>
<a name="IDX924"></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>
<p><code>load(affine)</code> loads this function.
</p>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Package-affine">Package affine</a>
</div>
</p></dd></dl>

<p><a name="fast_005fcentral_005felements"></a>
<a name="Item_003a-fast_005fcentral_005felements"></a>
</p><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="IDX925"></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><pre class="example">set_up_dot_simplifications ([y.x + x.y], 3);
fast_central_elements ([x, y], 2);
[y.y, x.x];
</pre>
<p><code>load(affine)</code> loads this function.
</p>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Package-affine">Package affine</a>
</div>
</p></dd></dl>

<p><a name="check_005foverlaps"></a>
<a name="Item_003a-check_005foverlaps"></a>
</p><dl>
<dt><u>Function:</u> <b>check_overlaps</b><i> (<var>n</var>, <var>add_to_simps</var>)</i>
<a name="IDX926"></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>
<p><code>load(affine)</code> loads this function.
</p>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Package-affine">Package affine</a>
</div>
</p></dd></dl>

<p><a name="mono"></a>
<a name="Item_003a-mono"></a>
</p><dl>
<dt><u>Function:</u> <b>mono</b><i> ([<var>x_1</var>, ..., <var>x_n</var>], <var>n</var>)</i>
<a name="IDX927"></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>
<p><code>load(affine)</code> loads this function.
</p>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Package-affine">Package affine</a>
</div>
</p></dd></dl>

<p><a name="monomial_005fdimensions"></a>
<a name="Item_003a-monomial_005fdimensions"></a>
</p><dl>
<dt><u>Function:</u> <b>monomial_dimensions</b><i> (<var>n</var>)</i>
<a name="IDX928"></a>
</dt>
<dd><p>Compute the Hilbert series through degree <var>n</var> for the current algebra.
</p>
<p><code>load(affine)</code> loads this function.
</p>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Package-affine">Package affine</a>
</div>
</p></dd></dl>

<p><a name="extract_005flinear_005fequations"></a>
<a name="Item_003a-extract_005flinear_005fequations"></a>
</p><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="IDX929"></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>
<p><code>load(affine)</code> loads this function.
</p>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Package-affine">Package affine</a>
</div>
</p></dd></dl>

<p><a name="list_005fnc_005fmonomials"></a>
<a name="Item_003a-list_005fnc_005fmonomials"></a>
</p><dl>
<dt><u>Function:</u> <b>list_nc_monomials</b><i> &nbsp;<br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>list_nc_monomials</tt> ([<var>p_1</var>, ..., <var>p_n</var>]) &nbsp;<br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>list_nc_monomials</tt> (<var>p</var>)</i>
<a name="IDX930"></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>
<p><code>load(affine)</code> loads this function.
</p>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Package-affine">Package affine</a>
</div>
</p></dd></dl>



<p><a name="all_005fdotsimp_005fdenoms"></a>
<a name="Item_003a-all_005fdotsimp_005fdenoms"></a>
</p><dl>
<dt><u>Option variable:</u> <b>all_dotsimp_denoms</b>
<a name="IDX931"></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>
<div class=categorybox>

<p>Categories:&nbsp;&nbsp;<a href="maxima_105.html#Category_003a-Package-affine">Package affine</a>
</div>
</p>
</dd></dl>




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