Syzygies over the polytope semiring

Manjunath Madhusudan

摘要

Abstract > Tropical geometry and its applications indicate a ‘theory of syzygies’ over polytope semirings. Taking cue from this indication, we study a notion of syzygies over the polytope semiring. We begin our exploration with the concept of Newton basis, an analogue of Gr?bner basis that captures the image of an ideal under the Newton polytope map. The image <mat:math xmlns:mat="http://www.w3.org/1998/Math/MathML" display="inline" altimg="urn:x-wiley:00246107:media:jlms12065:jlms12065-math-0001" xmlns:wiley="http://www.wiley.com/namespaces/wiley/wiley" wiley:location="equation/jlms12065-math-0001.png"> <mat:mrow> <mat:mi>New</mat:mi> <mat:mo stretchy="false">(</mat:mo> <mat:mi>I</mat:mi> <mat:mo stretchy="false">)</mat:mo> </mat:mrow> </mat:math> of a graded ideal <mat:math xmlns:mat="http://www.w3.org/1998/Math/MathML" display="inline" altimg="urn:x-wiley:00246107:media:jlms12065:jlms12065-math-0002" xmlns:wiley="http://www.wiley.com/namespaces/wiley/wiley" wiley:location="equation/jlms12065-math-0002.png"> <mat:mi>I</mat:mi> </mat:math> under the Newton polytope map is a graded sub‐semimodule of the polytope semiring. Analogous to the Hilbert series, we define the notion of Newton–Hilbert series that encodes the rank of each graded piece of <mat:math xmlns:mat="http://www.w3.org/1998/Math/MathML" display="inline" altimg="urn:x-wiley:00246107:media:jlms12065:jlms12065-math-0003" xmlns:wiley="http://www.wiley.com/namespaces/wiley/wiley" wiley:location="equation/jlms12065-math-0003.png"> <mat:mrow> <mat:mi>New</mat:mi> <mat:mo stretchy="false">(</mat:mo> <mat:mi>I</mat:mi> <mat:mo stretchy="false">)</mat:mo> </mat:mrow> </mat:math> . We prove the rationality of the Newton–Hilbert series for sub‐semimodules that satisfy a property analogous to Cohen‐Macaulayness. We define the notions of regular sequence of polytopes and syzygies of polytopes. We show an analogue of the Koszul property characterizing the syzygies of a regular sequence of polytopes. </abstract> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> <div class="translation abstracttxt"> <span class="zhankaihshouqi fivelineshidden" id="abstract"> <span>机译：</span><abstract xmlns=”http://www.wiley.com/namespaces/wiley“type=“main”><title type=“main”“>摘要>热带几何学及其应用表明了多面体半环上的“syzygies理论”。根据这一指示，我们研究了多面体半环上的syzygies概念。我们从牛顿基的概念开始探索，这是一种类似于Gr？bner基的概念，它捕捉了牛顿多面体映射下理想的图像<mat:math xmlns:mat=”http://www.w3.org/1998/Math/MathML“display=“inline”altimg=“urn:x-wiley:00246107:media:jlms12065:jlms12065-math-0001”xmlns:wiley=”http://www.wiley.com/namespaces/wiley/wiley“wiley:location=”等式/jlms12065-math-0001。png“><mat:mrow><mat:mi>新的</mat:mi><mat:mo stretchy=“false”>（</mat:mo><mat:mi>I</mat:mi><mat:mo stretchy=“false”>）</mat:mo></mat:mrow></mat:math><math xmlns:mat=”http://www.w3.org/1998/Math/MathML“display=“inline”altimg=“urn:x-wiley:00246107:media:jlms12065:jlms12065-math-0002”xmlns:wiley=”http://www.wiley.com/namespaces/wiley/wiley“wiley:location=”等式/jlms12065-math-0002。png“><mat:mi>I</mat:mi></mat:math>在牛顿多面体映射下是多面体半环的一个分次半模。与希尔伯特级数类似，我们定义了牛顿-希尔伯特级数的概念，它对<mat:math xmlns:mat=”的每个分次块的秩进行编码http://www.w3.org/1998/Math/MathML“display=“inline”altimg=“urn:x-wiley:00246107:media:jlms12065:jlms12065-math-0003”xmlns:wiley=”http://www.wiley.com/namespaces/wiley/wiley“wiley:location=”等式/jlms12065-math-0003。png“><mat:mrow><mat:mi>新的</mat:mi><mat:mo stretchy=“false”>（</mat:mo><mat:mi>I</mat:mi><mat:mo stretchy=“false”>）</mat:mo></mat:mrow></mat:math>。对于满足类似于Cohen-Macaulaynes性质的子半模，我们证明了牛顿-希尔伯特级数的合理性。我们定义了多面体的正则序列和多面体的syzygies的概念。我们展示了Koszul性质的一个类似物，该性质刻画了一个规则多面体序列的syzygies </abstract> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> </div> <div class="record"> <h2 class="all_title" id="enpatent33" >著录项</h2> <ul> <li> <span class="lefttit">来源</span> <div style="width: 86%;vertical-align: text-top;display: inline-block;"> <a href='/journal-foreign-31023/'>《The Journal of the London Mathematical Society》</a> <b style="margin: 0 2px;">|</b><span>2017年第2期</span><b style="margin: 0 2px;">|</b><span>共19页</span> </div> </li> <li> <div class="author"> <span class="lefttit">作者</span> <p id="fAuthorthree" class="threelineshidden zhankaihshouqi"> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Manjunath Madhusudan&option=202" target="_blank" rel="nofollow">Manjunath Madhusudan;</a> </p> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zkzz" style="display: none;">展开▼</span> </div> </li> <li> <div style="display: flex;"> <span class="lefttit">作者单位</span> <div style="position: relative;margin-left: 3px;max-width: 639px;"> <div class="threelineshidden zhankaihshouqi" id="fOrgthree"> <p>Department of MathematicsIndian Institute of Technology BombayPowai Mumbai 400076 Maharashtra India;</p> </div> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zhdw" style="display: none;">展开▼</span> </div> </div> </li> <li > <span class="lefttit">收录信息</span> <span style="width: 86%;vertical-align: text-top;display: inline-block;"></span> </li> <li> <span class="lefttit">原文格式</span> <span>PDF</span> </li> <li> <span class="lefttit">正文语种</span> <span>eng</span> </li> <li> <span class="lefttit">中图分类</span> <span><a href="https://www.zhangqiaokeyan.com/clc/156.html" title="数学">数学;</a></span> </li> <li class="antistop"> <span class="lefttit">关键词</span> <p style="width: 86%;vertical-align: text-top;"> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=13D02&option=203" rel="nofollow">13D02;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=14T05&option=203" rel="nofollow">14T05;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=16E05&option=203" rel="nofollow">16E05;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=16Y60&option=203" rel="nofollow">16Y60;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=52B11 (primary)&option=203" rel="nofollow">52B11 (primary);</a> </p> <div class="translation"> 机译：13d02;14T05;16E05;16Y60;52B11（初级）; 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