+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Equidimensional Decomposition · AlgebraicSolving.jl</title><meta name="title" content="Equidimensional Decomposition · AlgebraicSolving.jl"/><meta property="og:title" content="Equidimensional Decomposition · AlgebraicSolving.jl"/><meta property="twitter:title" content="Equidimensional Decomposition · AlgebraicSolving.jl"/><meta name="description" content="Documentation for AlgebraicSolving.jl."/><meta property="og:description" content="Documentation for AlgebraicSolving.jl."/><meta property="twitter:description" content="Documentation for AlgebraicSolving.jl."/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" 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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Algorithms</a></li><li class="is-active"><a href>Equidimensional Decomposition</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Equidimensional Decomposition</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/ederc/AlgebraicSolving.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/ederc/AlgebraicSolving.jl/blob/main/docs/src/decomposition.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><ul><li><a href="#Equidimensional-Decomposition">Equidimensional Decomposition</a></li><li class="no-marker"><ul><li><a href="#Introduction">Introduction</a></li><li><a href="#Functionality">Functionality</a></li></ul></li></ul><h1 id="Equidimensional-Decomposition"><a class="docs-heading-anchor" href="#Equidimensional-Decomposition">Equidimensional Decomposition</a><a id="Equidimensional-Decomposition-1"></a><a class="docs-heading-anchor-permalink" href="#Equidimensional-Decomposition" title="Permalink"></a></h1><h2 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h2><p>AlgebraicSolving allows to compute equidimensional decompositions of polynomial ideals. This is to be understood in a geometric sense, i.e. given a polynomial ideal <span>$I$</span> it computes ideals <span>$I_1,\dots,I_k$</span> s.t. <span>$V(I)=\bigcup_{i=1}^{k} V(I_j)$</span> and such that each <span>$V(I_j)$</span> is equidimensional.</p><p>The implemented algorithm is the one given in <a href="https://arxiv.org/abs/2409.17785">this paper</a>.</p><h2 id="Functionality"><a class="docs-heading-anchor" href="#Functionality">Functionality</a><a id="Functionality-1"></a><a class="docs-heading-anchor-permalink" href="#Functionality" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="AlgebraicSolving.equidimensional_decomposition-Union{Tuple{Ideal{T}}, Tuple{T}} where T<:MPolyRingElem" href="#AlgebraicSolving.equidimensional_decomposition-Union{Tuple{Ideal{T}}, Tuple{T}} where T<:MPolyRingElem"><code>AlgebraicSolving.equidimensional_decomposition</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><p>function equidimensional<em>decomposition(I::Ideal{T}, info</em>level::Int=0) where {T <: MPolyRingElem}</p><p>Given a polynomial ideal <code>I</code>, return a list of ideals <code>dec</code> s.t. each ideal in <code>dec</code> is equidimensional (i.e. has minimal primes only of one fixed dimension) and s.t. the radical of <code>I</code> equals the intersection of the radicals of the ideals in <code>dec</code>.</p><p><strong>Note</strong>: At the moment only ground fields of characteristic <code>p</code>, <code>p</code> prime, <code>p < 2^{31}</code> are supported.</p><p><strong>Arguments</strong></p><ul><li><code>I::Ideal{T} where T <: MpolyElem</code>: input ideal.</li><li><code>info_level::Int=0</code>: info level printout: off (<code>0</code>, default), computational details (<code>1</code>)</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia> using AlgebraicSolving
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