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<!DOCTYPE html> <html lang="en"> <head> <title>genKer -- basis of the kernel of a linear expression</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="differential algebra" href="index.html">DiffAlg</a> :: <a title="basis of the kernel of a linear expression" href="_gen__Ker.html">genKer</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_homogenize_lp__Diff__Alg__Element_rp.html">next</a> | <a href="_gen__Im.html">previous</a> | <a href="_homogenize_lp__Diff__Alg__Element_rp.html">forward</a> | <a href="_gen__Im.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>genKer -- basis of the kernel of a linear expression</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">gerKer(expr,var)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">expr</span>, <span>an instance of the type <a title="the class of all differential forms and vector fields" href="___Diff__Alg__Element.html">DiffAlgElement</a></span>, an expression linear in the variable <span class="tt">var</span></span></li> <li><span><span class="tt">var</span>, <span>an instance of the type <a title="the class of all differential forms and vector fields" href="___Diff__Alg__Element.html">DiffAlgElement</a></span>, this is the variable of the linear expression, it must have free and linear scalar coefficients</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, basis of the kernel of the linear expression</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <p>This routine returns a basis of the kernel of <span class="tt">expr</span>, an homogeneous expression linear in <span class="tt">var</span>.</p> <br> <p>In the case of a non-homogeneous linear expression, this routine returns a pair having in the first coordinate a basis of the kernel of the associated homogeneous linear expression and in the second coordinate a particular solution.</p> <br> <p>In the first example, we compute a basis of projective differential 1-forms in projective 3-space with polynomial coefficients of degree 1. Then, we define a random rational differential form of type <span class="tt">(1,1)</span> and compute its tangent directions using the generic projective form defined before.</p> <br> <p>In the second example, we compute a particular solution of a non-homogeneous linear expression.</p> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : h = newForm(4,1,1,"a") o1 = (a x + a x + a x + a x + a x )dx + (a x + a x + a x + a x 0 0 5 1 10 2 15 3 20 4 0 1 0 6 1 11 2 16 3 ------------------------------------------------------------------------ + a x )dx + (a x + a x + a x + a x + a x )dx + (a x + a x + 21 4 1 2 0 7 1 12 2 17 3 22 4 2 3 0 8 1 ------------------------------------------------------------------------ a x + a x + a x )dx + (a x + a x + a x + a x + a x )dx 13 2 18 3 23 4 3 4 0 9 1 14 2 19 3 24 4 4 o1 : DiffAlgForm</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : R = radial 4 o2 = x ax + x ax + x ax + x ax + x ax 0 0 1 1 2 2 3 3 4 4 o2 : DiffAlgField</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : T = genKer(R _ h,h) o3 = {x dx - x dx , x dx - x dx , x dx - x dx , x dx - x dx , x dx - 1 0 0 1 2 0 0 2 2 1 1 2 3 0 0 3 3 1 ------------------------------------------------------------------------ x dx , x dx - x dx , x dx - x dx , x dx - x dx , x dx - x dx , x dx 1 3 3 2 2 3 4 0 0 4 4 1 1 4 4 2 2 4 4 3 ------------------------------------------------------------------------ - x dx } 3 4 o3 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : H = linearComb(T,"a") o4 = (a x + a x + a x + a x )dx + (- a x + a x + a x + a x )dx + (- 0 1 1 2 3 3 6 4 0 0 0 2 2 4 3 7 4 1 ------------------------------------------------------------------------ a x - a x + a x + a x )dx + (- a x - a x - a x + a x )dx + (- 1 0 2 1 5 3 8 4 2 3 0 4 1 5 2 9 4 3 ------------------------------------------------------------------------ a x - a x - a x - a x )dx 6 0 7 1 8 2 9 3 4 o4 : DiffAlgForm</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : w = random logarithmicForm(4,{1,1},"a", Projective => true) o5 = (49x + 77x + 14x - 28x )dx + (- 49x + 140x + 70x - 140x )dx + (- 1 2 3 4 0 0 2 3 4 1 ------------------------------------------------------------------------ 77x - 140x + 70x - 140x )dx + (- 14x - 70x - 70x )dx + (28x + 0 1 3 4 2 0 1 2 3 0 ------------------------------------------------------------------------ 140x + 140x )dx 1 2 4 o5 : DiffAlgForm</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : genKer(w ^ (diff H) + (diff w) ^ H,H) 1 1 1 o6 = {(x + x )dx - x dx - x dx , - -x dx + (-x + x )dx - x dx , -x dx 1 2 0 0 1 0 2 5 1 0 5 0 2 1 1 2 5 3 0 ------------------------------------------------------------------------ 1 1 + x dx + x dx + (- -x - x - x )dx , (x - -x + x )dx - x dx + 3 1 3 2 5 0 1 2 3 1 2 3 4 0 0 1 ------------------------------------------------------------------------ 1 11 11 1 1 7 -x dx - x dx , - --x dx + (--x - -x + x )dx + -x dx - x dx , (--x 2 0 3 0 4 20 1 0 20 0 2 3 4 1 2 1 3 1 4 20 1 ------------------------------------------------------------------------ 1 7 1 1 1 + --x )dx + (- --x + -x )dx + x dx + (- --x - -x )dx - x dx , - 10 3 0 20 0 2 3 1 4 2 10 0 2 1 3 2 4 ------------------------------------------------------------------------ 11 11 --x dx - x dx + (--x + x + x )dx - x dx } 20 3 0 3 1 20 0 1 4 3 3 4 o6 : List</code></pre> </td> </tr> </table> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i7 : w1 = random newForm(4,1,1,"a");</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : w2 = random newForm(4,1,1,"a");</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : w3 = w1 ^ w2;</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : h = newForm(4,1,1,"a");</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : last genKer(w1 ^ h - w3,h) 7 9 5 15 13 3 5 o11 = (- -x - x - -x + x + -x )dx + (--x - 5x - --x - -x + -x )dx + 4 0 1 4 2 3 4 4 0 2 0 1 2 2 4 3 2 4 1 ----------------------------------------------------------------------- 25 7 5 3 11 1 3 (- --x - -x - -x + -x - --x )dx + (-x - 4x - -x - 4x - 2x )dx 4 0 2 1 4 2 2 3 4 4 2 4 0 1 4 2 3 4 3 ----------------------------------------------------------------------- 11 3 + (--x - -x - 2x + 2x )dx 4 0 2 1 2 3 4 o11 : DiffAlgForm</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="a basis of the image of a linear expression" href="_gen__Im.html">genIm</a> -- a basis of the image of a linear expression</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">genKer</span>:</h2> <ul> <li><kbd>genKer(DiffAlgElement,DiffAlgElement)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="basis of the kernel of a linear expression" href="_gen__Ker.html">genKer</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">DiffAlg.m2:916:0</span>.</p> </div> </div> </div> </body> </html>
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