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___Replacement.html
<!DOCTYPE html> <html lang="en"> <head> <title>Replacement -- When changing coordinates, whether to replace variables by general degree 1 forms, binomials, etc.</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="Obtain random points in a variety" href="index.html">RandomPoints</a> :: <a title="When changing coordinates, whether to replace variables by general degree 1 forms, binomials, etc." href="___Replacement.html">Replacement</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> next | <a href="_random__Points_lp..._cm__Strategy_eq_gt..._rp.html">previous</a> | forward | <a href="_random__Points_lp..._cm__Strategy_eq_gt..._rp.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>Replacement -- When changing coordinates, whether to replace variables by general degree 1 forms, binomials, etc.</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">Replacement => Full</code></dd> <dd><code class="language-macaulay2">Replacement => Monomial</code></dd> <dd><code class="language-macaulay2">Replacement => Binomial</code></dd> <dd><code class="language-macaulay2">Replacement => Trinomial</code></dd> </dl> </li> </ul> <div> <h2>Description</h2> <div> <p>When calling various functions, setting <span class="tt">Replacement => Full</span> will mean that coordinates are changed to a general degree 1 form. If <span class="tt">Replacement => Binomial</span>, the coordinates are only changed to binomials, which can be much faster for certain applications. Other options include <span class="tt">Replacement => Monomial</span> and <span class="tt">Replacement => Trinomial</span>.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = ZZ/101[a,b,c,d,e];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : randomCoordinateChange(R, Replacement=>Monomial) ZZ o2 = map (R, ---[a..e], {a, d, e, b, c}) 101 ZZ o2 : RingMap R <-- ---[a..e] 101</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : randomCoordinateChange(R, Replacement=>Binomial) ZZ o3 = map (R, ---[a..e], {- 16a + d, -12c, - 24d + e, a + 19d, b + 21c}) 101 ZZ o3 : RingMap R <-- ---[a..e] 101</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : randomCoordinateChange(R, Replacement=>Trinomial) ZZ o4 = map (R, ---[a..e], {22b + 45d + e, 47b + 40d, 11b + c + 36d, - 11a + d + 43e, a - 16b - 48d}) 101 ZZ o4 : RingMap R <-- ---[a..e] 101</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : randomCoordinateChange(R, Replacement=>Full) ZZ o5 = map (R, ---[a..e], {- 10a + 30b - 18c + 39d + 27e, 24a - 30b - 48c - 15d + 39e, 29a - 47b + 15c - 37d - 13e, - 22a + 32b - 9c - 32d - 20e, 33b - 49c - 33d - 19e}) 101 ZZ o5 : RingMap R <-- ---[a..e] 101</code></pre> </td> </tr> </table> <div> <p>If <span class="tt">Homogeneous => false</span> in these cases, then there will be constant terms, and we view $mx + b$ as a monomial.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i6 : S = ZZ/103[x,y,z,u,v];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : randomCoordinateChange(S, Replacement => Monomial, Homogeneous => false) ZZ o7 = map (S, ---[x..z, u..v], {z + 4, y + 9, x + 13, v - 28, u - 41}) 103 ZZ o7 : RingMap S <-- ---[x..z, u..v] 103</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : randomCoordinateChange(S, Replacement => Binomial, Homogeneous => false) ZZ o8 = map (S, ---[x..z, u..v], {- 32y + 3, - 32y + u - 24, z - 30u + 16, x - 43v - 4, - 37x + v - 11}) 103 ZZ o8 : RingMap S <-- ---[x..z, u..v] 103</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : randomCoordinateChange(S, Replacement => Trinomial, Homogeneous => false) ZZ o9 = map (S, ---[x..z, u..v], {- 14u + 30v + 7, - 20y + 41z + 28, - 19x + 42y + 49, 30x + 48v - 50, 8x + y - 31z - 43}) 103 ZZ o9 : RingMap S <-- ---[x..z, u..v] 103</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : randomCoordinateChange(S, Replacement => Full, Homogeneous => false) ZZ o10 = map (S, ---[x..z, u..v], {6x - 11y - 35z + 28u - 31v - 30, 30x + 4y + 22z + 5u - 22v + 44, 19x - 49z - 30u + 47v + 20, 5x - 39y - 35z - 30u + 42v + 26, 23x - 39y - 25z + 44u - 41v - 23}) 103 ZZ o10 : RingMap S <-- ---[x..z, u..v] 103</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="produce linear automorphism of the ring" href="_random__Coordinate__Change.html">randomCoordinateChange</a> -- produce linear automorphism of the ring</span></li> </ul> </div> <div> <div> <h2>Functions with optional argument named <span class="tt">Replacement</span>:</h2> <ul> <li><kbd>dimViaBezout(...,Replacement=>...)</kbd></li> <li><kbd>extendIdealByNonZeroMinor(...,Replacement=>...)</kbd></li> <li><kbd>findANonZeroMinor(...,Replacement=>...)</kbd></li> <li><kbd>genericProjection(...,Replacement=>...)</kbd></li> <li><kbd>projectionToHypersurface(...,Replacement=>...)</kbd></li> <li><kbd>randomCoordinateChange(...,Replacement=>...)</kbd></li> <li><kbd>randomPoints(...,Replacement=>...)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="When changing coordinates, whether to replace variables by general degree 1 forms, binomials, etc." href="___Replacement.html">Replacement</a> is <span>a <a title="the class of all symbols" href="../../Macaulay2Doc/html/___Symbol.html">symbol</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">RandomPoints.m2:1826:0</span>.</p> </div> </div> </div> </body> </html>