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”>抽象</ title> >热带几何形状及其应用表明了“理论” syzygies'过多的雾化。从这种迹象中提出提示，我们研究了对多晶石精彩的Syzygies的概念。我们开始探索牛顿基础的概念，这是GR的模拟的基础，捕获牛顿多特素映射下的理想形象。图像<mat：math xml：mat =“http://www.w3.org/1998/math/mathml”display =“内联”altimg =“urn：x-wiley：00246107：媒体：jlms12065：jlms12065-数学-0001“XMLNS：Wiley =”http://www.wiley.com/namespaces/wiley/wiley“wiley：location =”等式/ jlms12065-math-0001.png“> <mat：mrow> <mat：mi> New </ mat：mi> <mat：mo stractry =“false”>（</ mat：mo> <mat：mi> i </ mat：mi> <mat：mo stractry =“假”>）</ mat ：mo> </ mat：mrow> </ mat：数学>分级理想<mat：math xmlns：mat =“http://www.w3.org/1998/math/mathml”display =“内联”altimg =“URN：X-Wiley：00246107：媒体：jlms12065：jlms12065-math-0002”xmlns：wiley.com/namespaces/wiley/wiley“wiley：location =”等式/ jlms12065- math-0002.png“> <mat：mi> i </ mat：mi> </ mat：math>在牛顿多托映射下是多托曲面的分级次半模。类似于希尔伯特系列，我们定义了编码每个分级<mat：math xmlns：mat =“http://www.w3.org/1998/math/mathml”显示的牛顿 - 希尔伯特系列的概念=“内联”Altimg =“URN：X-Wiley：00246107：媒体：jlms12065：jlms12065-math-0003”xmlns：wiley.com/namespaces/wiley/wiley“wiley：location = “等式/ jlms12065-math-0003.png”> <mat：mrow> <mat：mi> new </ mat：mi> <mat：mo stractry =“false”>（</ mat：mo> <mat：mi > I </ mat：mi> <mat：mo stractry =“false”>）</ mat：mo> </ mat：mrow> </ mat：math>。我们证明了牛顿 - 希尔伯特系列的合理性，以满足类似于科恩 - 澳门的属性的亚半代理。我们定义了常规序列的多核糖和多粒子的Syzygies概念。我们展示了Koszul属性的类似物，其特征在于常规序列的多粒子的Syzygies。 </ p> </ 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）; </div> </li> </ul> </div> </div> <div class="literature cardcommon"> <div class="similarity "> <h3 class="all_title" id="enpatent66">相似文献</h3> <div class="similaritytab clearfix"> <ul> <li class="active" >外文文献</li> <li >中文文献</li> <li >专利</li> </ul> </div> <div class="similarity_details"> <ul > <li> <div> <b>1. </b><a class="enjiyixqcontent" href="/journal-foreign-detail/0704024866727.html">Syzygies over the polytope semiring</a> <b>[J]</b> . <span> <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" class="tuijian_auth tuijian_authcolor">Manjunath Madhusudan </a> <a 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