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<?xml version="1.0" encoding="UTF-8" ?>
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<html xmlns="http://www.w3.org/1999/xhtml"
xmlns:xlink="http://www.w3.org/1999/xlink" >
<head>
<title>Section1.8</title>
<link rel="stylesheet" type="text/css" href="graphicstyle.css" />
<script type="text/javascript" src="bookax1.js" />
</head>
<body>
<a href="book-contents.xhtml" style="margin-right: 10px;">Book Contents</a><a href="section-1.7.xhtml" style="margin-right: 10px;">Previous Section 1.7 Writing Your Own Functions</a><a href="section-1.9.xhtml" style="margin-right: 10px;">Next Section 1.9 Limits</a>
<a href="book-index.xhtml">Book Index</a><div class="section" id="sec-1.8">
<h2 class="sectiontitle">1.8 Polynomials</h2>
<a name="ugIntroVariables" class="label"/>
<p>Polynomials are the commonly used algebraic types in symbolic
computation. <span class="index">polynomial</span><a name="chapter-1-67"/> Interactive users of Axiom
generally only see one type of polynomial: <span class="teletype">Polynomial(R)</span>.
This type represents polynomials in any number of unspecified
variables over a particular coefficient domain <math xmlns="&mathml;" mathsize="big"><mstyle><mi>R</mi></mstyle></math>. This type
represents its coefficients <span style="font-style: italic;"> sparsely</span>: only terms with non-zero
coefficients are represented.
</p>
<p>In building applications, many other kinds of polynomial
representations are useful. Polynomials may have one variable or
multiple variables, the variables can be named or unnamed, the
coefficients can be stored sparsely or densely. So-called
``distributed multivariate polynomials'' store polynomials as
coefficients paired with vectors of exponents. This type is
particularly efficient for use in algorithms for solving systems of
non-linear polynomial equations.
</p>
<p>The polynomial constructor most familiar to the interactive user
is <span class="teletype">Polynomial</span>.
</p>
<div id="spadComm1-127" class="spadComm" >
<form id="formComm1-127" action="javascript:makeRequest('1-127');" >
<input id="comm1-127" type="text" class="command" style="width: 16em;" value="(x**2 - x*y**3 +3*y)**2" />
</form>
<span id="commSav1-127" class="commSav" >(x**2 - x*y**3 +3*y)**2</span>
<div id="mathAns1-127" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo></mo><mrow><msup><mi>y</mi><mn>6</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>6</mn><mo></mo><mi>x</mi><mo></mo><mrow><msup><mi>y</mi><mn>4</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo></mo><mrow><msup><mi>y</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>9</mn><mo></mo><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>6</mn><mo></mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo></mo><mi>y</mi></mrow><mo>+</mo><mrow><msup><mi>x</mi><mn>4</mn></msup></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Polynomial Integer
</div>
<p>If you wish to restrict the variables used,
<span class="teletype">UnivariatePolynomial</span> provides polynomials in one variable.
</p>
<div id="spadComm1-128" class="spadComm" >
<form id="formComm1-128" action="javascript:makeRequest('1-128');" >
<input id="comm1-128" type="text" class="command" style="width: 26em;" value="p: UP(x,INT) := (3*x-1)**2 * (2*x + 8)" />
</form>
<span id="commSav1-128" class="commSav" >p: UP(x,INT) := (3*x-1)**2 * (2*x + 8)</span>
<div id="mathAns1-128" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mn>18</mn><mo></mo><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>60</mn><mo></mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>46</mn><mo></mo><mi>x</mi></mrow><mo>+</mo><mn>8</mn></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: UnivariatePolynomial(x,Integer)
</div>
<p>The constructor <span class="teletype">MultivariatePolynomial</span> provides polynomials
in one or more specified variables.
</p>
<div id="spadComm1-129" class="spadComm" >
<form id="formComm1-129" action="javascript:makeRequest('1-129');" >
<input id="comm1-129" type="text" class="command" style="width: 29em;" value="m: MPOLY([x,y],INT) := (x**2-x*y**3+3*y)**2" />
</form>
<span id="commSav1-129" class="commSav" >m: MPOLY([x,y],INT) := (x**2-x*y**3+3*y)**2</span>
<div id="mathAns1-129" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><msup><mi>x</mi><mn>4</mn></msup></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><msup><mi>y</mi><mn>3</mn></msup></mrow><mo></mo><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>6</mn></msup></mrow><mo>+</mo><mrow><mn>6</mn><mo></mo><mi>y</mi></mrow><mo>)</mo></mrow><mo></mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>6</mn><mo></mo><mrow><msup><mi>y</mi><mn>4</mn></msup></mrow><mo></mo><mi>x</mi></mrow><mo>+</mo><mrow><mn>9</mn><mo></mo><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: MultivariatePolynomial([x,y],Integer)
</div>
<p>You can change the way the polynomial appears by modifying the variable
ordering in the explicit list.
</p>
<div id="spadComm1-130" class="spadComm" >
<form id="formComm1-130" action="javascript:makeRequest('1-130');" >
<input id="comm1-130" type="text" class="command" style="width: 14em;" value="m :: MPOLY([y,x],INT)" />
</form>
<span id="commSav1-130" class="commSav" >m :: MPOLY([y,x],INT)</span>
<div id="mathAns1-130" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo></mo><mrow><msup><mi>y</mi><mn>6</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>6</mn><mo></mo><mi>x</mi><mo></mo><mrow><msup><mi>y</mi><mn>4</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo></mo><mrow><msup><mi>y</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>9</mn><mo></mo><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>6</mn><mo></mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo></mo><mi>y</mi></mrow><mo>+</mo><mrow><msup><mi>x</mi><mn>4</mn></msup></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: MultivariatePolynomial([y,x],Integer)
</div>
<p>The constructor <span class="teletype">DistributedMultivariatePolynomial</span> provides
polynomials in one or more specified variables with the monomials
ordered lexicographically.
</p>
<div id="spadComm1-131" class="spadComm" >
<form id="formComm1-131" action="javascript:makeRequest('1-131');" >
<input id="comm1-131" type="text" class="command" style="width: 13em;" value="m :: DMP([y,x],INT)" />
</form>
<span id="commSav1-131" class="commSav" >m :: DMP([y,x],INT)</span>
<div id="mathAns1-131" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mrow><msup><mi>y</mi><mn>6</mn></msup></mrow><mo></mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>6</mn><mo></mo><mrow><msup><mi>y</mi><mn>4</mn></msup></mrow><mo></mo><mi>x</mi></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><msup><mi>y</mi><mn>3</mn></msup></mrow><mo></mo><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>9</mn><mo></mo><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>6</mn><mo></mo><mi>y</mi><mo></mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><msup><mi>x</mi><mn>4</mn></msup></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: DistributedMultivariatePolynomial([y,x],Integer)
</div>
<p>The constructor
<span class="teletype">HomogeneousDistributedMultivariatePolynomial</span> is similar
except that the monomials are ordered by total order refined by
reverse lexicographic order.
</p>
<div id="spadComm1-132" class="spadComm" >
<form id="formComm1-132" action="javascript:makeRequest('1-132');" >
<input id="comm1-132" type="text" class="command" style="width: 14em;" value="m :: HDMP([y,x],INT)" />
</form>
<span id="commSav1-132" class="commSav" >m :: HDMP([y,x],INT)</span>
<div id="mathAns1-132" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mrow><msup><mi>y</mi><mn>6</mn></msup></mrow><mo></mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><msup><mi>y</mi><mn>3</mn></msup></mrow><mo></mo><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>6</mn><mo></mo><mrow><msup><mi>y</mi><mn>4</mn></msup></mrow><mo></mo><mi>x</mi></mrow><mo>+</mo><mrow><msup><mi>x</mi><mn>4</mn></msup></mrow><mo>+</mo><mrow><mn>6</mn><mo></mo><mi>y</mi><mo></mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>9</mn><mo></mo><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: HomogeneousDistributedMultivariatePolynomial([y,x],Integer)
</div>
<p>More generally, the domain constructor
<span class="teletype">GeneralDistributedMultivariatePolynomial</span> allows the user to
provide an arbitrary predicate to define his own term ordering. These
last three constructors are typically used in Gröbner basis
applications and
when a flat (that is, non-recursive) display is wanted and the term
ordering is critical for controlling the computation.
</p>
</div><a href="book-contents.xhtml" style="margin-right: 10px;">Book Contents</a>
<a href="section-1.7.xhtml" style="margin-right: 10px;">Previous Section 1.7 Writing Your Own Functions</a><a href="section-1.9.xhtml" style="margin-right: 10px;">Next Section 1.9 Limits</a>
<a href="book-index.xhtml">Book Index</a></body>
</html>
|
