fa مطالعه و دسته‌بندی برخی از کلاسهای ایده‌آلهای پلی‌ماترویدال ریاضی Mat مقاله مستقل Original Manuscript <div><span dir="RTL" lang="AR-SA" style="font-size:10.0pt"><span style="line-height:107%"><span b="" nazanin="" style="font-family:">در این مقاله رده ایده‌آل‌های پلی‌ماترویدال مورد مطالعه قرار گرفته‌اند. به ویژه نشان می‌دهیم که هر ایده‌آل پلی‌ماترویدال، تابع تجزیۀ منظم<a href="#_ftn1" name="_ftnref1" title=""><span class="MsoFootnoteReference" style="vertical-align:super"><span class="MsoFootnoteReference" style="vertical-align:super"><span style="font-size:10.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">[1]</span></span></span></span></span></a> دارد و لذا می‌توانیم تحلیل خطی دقیق آن را بیان کنیم. همچنین ایده‌آل‌های پلی‌ماترویدال عام را دسته‌بندی می‌کنیم. در نهایت به دسته‌بندی ایده‌آل‌های یک‌جمله‌ای که همه توان‌هایشان پلی‌ماترویدال کوهن- مکالی تعمیم یافته هستند، می‌پردازیم.</span></span></span> <div>&nbsp; <div id="ftn1"></div></div> <br clear="all" > &nbsp;</div> <div></div> <span style="font-size:14px;"><span style="line-height:16.0pt"><b><span new="" roman="" style="font-family:" times="">Introduction</span></b></span></span><br> <span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt">Throughout this paper, we consider monomial ideals of the polynomial ring <span style="position:relative"><span style="top:6.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:102px; height:24px" ></span></span>&nbsp;over a filed<span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image004.gif" style="width:18px; height:18px" ></span></span>. We try to give some properties of the polymatroidal ideals, which are the special class of monomial ideals. Herzog and Takayama constructed explicit resolutions for all ideals with linear quotients which admit regular decomposition functions. They also shaw that this class contains all matroidal ideals. We generalize their result to the polymatroidal ideals. Therefore, we can give an explicit linear resolution for any polymatroidal ideal. We also characterize generic polymatroidal ideals. The author and Jafari [1] characterized generalized Cohen-Macaulay polymatroidal ideals. Finally, we characterize monomial ideals which all their powers are generalized Cohen-Macaulay polymatroidal ideals.</span></span></span><br> <span style="line-height:16.0pt"><b><span style="font-size:9.0pt">Material and methods</span></b></span><br> <span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt">A monomial ideal <span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image006.gif" style="width:14px; height:17px" ></span></span>&nbsp;is said to be polymatroidal, if it is single degree and for any two elements <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image008.gif" style="width:72px; height:21px" ></span></span>&nbsp;such that <span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image010.gif" style="width:123px; height:26px" ></span></span>&nbsp;there exists an index <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image012.gif" style="width:46px; height:21px" ></span></span>&nbsp;with <span style="position:relative"><span style="top:8.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image014.gif" style="width:128px; height:27px" ></span></span>&nbsp;such that<span style="position:relative"><span style="top:15.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image016.gif" style="width:69px; height:46px" ></span></span>.<span style="color:black"> In the case that the polymatroidal ideal <span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image018.gif" style="width:14px; height:18px" ></span></span>is squarefree, it is called matroidal. </span></span></span></span><br> <span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt">We know that the powers of a polymatroidal ideal are again polymatroidal and polymatroidal ideals have linear quotients. Therefore all powers of polymatroidal ideal have linear resolutions. </span></span></span><br> <span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt">Let <span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image018.gif" style="width:14px; height:18px" ></span></span>&nbsp;has linear quotients with the order <span style="position:relative"><span style="top:6.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image021.gif" style="width:54px; height:24px" ></span></span>&nbsp;of elements of<span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image023.gif" style="width:36px; height:22px" ></span></span>. We can associate a unique decomposition function, that is a function <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image025.gif" style="width:110px; height:22px" ></span></span>&nbsp;which maps a monomial <span style="position:relative"><span style="top:3.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image027.gif" style="width:14px; height:14px" ></span></span>&nbsp;to<span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image029.gif" style="width:18px; height:26px" ></span></span>, if <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image031.gif" style="width:14px; height:20px" ></span></span>&nbsp;is the smallest index such that <span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image033.gif" style="width:40px; height:26px" ></span></span>, where <span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image035.gif" style="width:98px; height:26px" ></span></span>. The decomposition function <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image025.gif" style="width:110px; height:22px" ></span></span>&nbsp;is called regular, if <span style="position:relative"><span style="top:6.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image038.gif" style="width:96px; height:24px" ></span></span>&nbsp;for all <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image040.gif" style="width:58px; height:22px" ></span></span>&nbsp;and<span style="position:relative"><span style="top:6.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image042.gif" style="width:60px; height:24px" ></span></span>.&nbsp; </span></span></span><br> <span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt">We show that any polymatroidal ideal has a regular decomposition function. Therefore we can give an explicit linear resolution for any polymatroidal ideal. By an example, we show that our result can not be extended to the weakly polymatroidal ideals even if they are generated in a single degree. </span></span></span><br> <span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt"><span style="color:black">Recall that, a monomial ideal <span style="position:relative"><span style="top:6.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image044.gif" style="width:100px; height:24px" ></span></span>&nbsp;is called generic if two distinct minimal generators <span style="position:relative"><span style="top:6.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image046.gif" style="width:18px; height:24px" ></span></span>&nbsp;and <span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image048.gif" style="width:22px; height:26px" ></span></span>&nbsp;have the same positive degree in some variable <span style="position:relative"><span style="top:6.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image050.gif" style="width:18px; height:24px" ></span></span>, there is a third generator <span style="position:relative"><span style="top:6.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image052.gif" style="width:20px; height:24px" ></span></span>&nbsp;which <span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image054.gif" style="width:102px; height:26px" ></span></span>&nbsp;and <span style="position:relative"><span style="top:15.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image056.gif" style="width:254px; height:48px" ></span></span>, where <span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image058.gif" style="width:78px; height:26px" ></span></span>&nbsp;is the least common multiple of <span style="position:relative"><span style="top:6.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image046.gif" style="width:18px; height:24px" ></span></span>&nbsp;and <span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image048.gif" style="width:22px; height:26px" ></span></span>.</span></span></span></span><br> <span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt"><span style="color:black">In the next result</span></span><span style="font-size:9.0pt">, we characterize generic polymatroidal ideals. <span style="color:black"></span></span></span></span><br> <span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt"><span style="color:black">A monomial ideal <span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image018.gif" style="width:14px; height:18px" ></span></span>&nbsp;is called generalized Cohen-Macaulay, whenever <span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image018.gif" style="width:14px; height:18px" ></span></span>&nbsp;is equidimensional and monomial localization <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image062.gif" style="width:34px; height:22px" ></span></span>&nbsp;is Cohen-Macaulay for all monomial prime ideals<span style="position:relative"><span style="top:3.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image064.gif" style="width:44px; height:18px" ></span></span>, where <span style="position:relative"><span style="top:3.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image066.gif" style="width:18px; height:14px" ></span></span>&nbsp;is unique homogenous maximal ideal of <span style="position:relative"><span style="top:3.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image068.gif" style="width:14px; height:18px" ></span></span>.</span></span><span style="font-size:9.0pt"></span></span></span><br> <span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt">Finally, we characterize monomial ideals which all their powers are generalized Cohen-Macaulay polymatroidal ideals.</span><span style="font-size:9.0pt"><span style="color:black"></span></span></span></span><br> <span style="line-height:16.0pt"><span style="text-autospace:none"><b><span style="font-size:9.0pt">Results and discussion</span></b></span></span><br> <span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt"><span style="color:black">For the first result, we show that any polymatroidal ideal has a regular decomposition function. So we have an explicit linear resolution of any polymatrodal ideal. </span></span><b><span style="font-size:9.0pt"></span></b></span></span><br> <span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt"><span style="color:black">In the next, we show that if<span style="position:relative"><span style="top:6.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image070.gif" style="width:113px; height:21px" ></span></span>is a fully supported</span></span> <span style="font-size:9.0pt"><span style="color:black">polymatroidal ideal generated in degree<span style="position:relative"><span style="top:3.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image072.gif" style="width:14px; height:18px" ></span></span>. Then <span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image018.gif" style="width:14px; height:18px" ></span></span>&nbsp;is generic if and only if <span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image018.gif" style="width:14px; height:18px" ></span></span>&nbsp;is either a complete intersection or<span style="position:relative"><span style="top:3.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image076.gif" style="width:38px; height:18px" ></span></span>.</span></span></span></span><br> <span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt">Finally, we prove that if<b> </b><span style="color:black">&nbsp;<span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image018.gif" style="width:14px; height:18px" ></span></span>&nbsp;is a fully supported monomial ideal in <span style="position:relative"><span style="top:6.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:102px; height:24px" ></span></span>&nbsp;and generated in degree<span style="position:relative"><span style="top:3.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image072.gif" style="width:14px; height:18px" ></span></span>. Then<span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image081.gif" style="width:18px; height:20px" ></span></span>&nbsp;is a generalized Cohen-Macaulay polymatroidal ideal for all <span style="position:relative"><span style="top:3.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image083.gif" style="width:41px; height:18px" ></span></span>&nbsp;if and only if <span style="position:relative"><span style="top:3.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image085.gif" style="width:72px; height:22px" ></span></span>where <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image087.gif" style="width:62px; height:22px" ></span></span>&nbsp;and <span style="position:relative"><span style="top:6.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image089.gif" style="width:125px; height:26px" ></span></span>&nbsp;for some integers <span style="position:relative"><span style="top:6.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image091.gif" style="width:41px; height:24px" ></span></span>&nbsp;and one of the following statements holds true:</span></span></span></span> <ol style="list-style-type:lower-alpha"> <li class="CxSpMiddle" style="margin-top:8px; margin-bottom:8px"><span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt"><span style="font-family:"Times New Roman",serif"></span></span><span style="font-size:9.0pt"><span style="position:relative"><span style="top:3.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image093.gif" style="width:14px; height:18px" ></span></span></span><span style="font-size:9.0pt"><span style="color:black">&nbsp;is a principal ideal.</span></span></span></span></li> <li class="CxSpMiddle" style="margin-top:8px; margin-bottom:8px"><span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt"><span style="font-family:"Times New Roman",serif"></span></span><span style="font-size:9.0pt"><span style="color:black"><span style="position:relative"><span style="top:3.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image093.gif" style="width:14px; height:18px" ></span></span>&nbsp;is a Veronese ideal.</span></span></span></span></li> <li class="CxSpMiddle" style="margin-top:8px; margin-bottom:8px"><span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt"><span style="font-family:"Times New Roman",serif"></span></span><span style="font-size:9.0pt"><span style="color:black"><span style="position:relative"><span style="top:6.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image096.gif" style="width:114px; height:23px" ></span></span>is equidimensional and <span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image098.gif" style="width:68px; height:22px" ></span></span>&nbsp;for all <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image100.gif" style="width:34px; height:20px" ></span></span>.</span></span></span></span></li> <li class="CxSpMiddle" style="margin-top:8px; margin-bottom:8px"><span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt"><span style="font-family:"Times New Roman",serif"></span></span><span style="font-size:9.0pt"><span style="color:black"><span style="position:relative"><span style="top:3.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image093.gif" style="width:14px; height:18px" ></span></span>&nbsp;is an unmixed matroidal ideal of degree 2.</span></span></span></span></li> </ol> <span style="line-height:16.0pt"><b><span style="font-size:9.0pt">Conclusion</span></b></span><br> <span style="line-height:16.0pt"><span style="font-size:9.0pt">The following conclusions were drawn from this research:</span></span> <ul> <li class="CxSpMiddle" style="margin-top:8px; margin-bottom:8px"><span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt"></span><span style="font-size:9.0pt">Any polymatroidal ideal has <span style="color:black">a regular decomposition function.</span></span></span></span></li> <li class="CxSpMiddle" style="margin-top:8px; margin-bottom:8px"><span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt"></span><span style="font-size:9.0pt">characterization of generic ideals.</span></span></span></li> <li class="CxSpMiddle" style="margin-top:8px; margin-bottom:8px"><span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:9.0pt"></span><span style="font-size:9.0pt">characterization <span style="color:black">of </span>monomial ideals which all their powers are generalized Cohen-Macaulay polymatroidal ideals.<span style="color:black"></span></span></span></span></li> </ul> ایده‌آلهای پلی‌ماترویدال, تابع تجزیه منظم, ایده‌آلهای عام , ایده‌آلهای کوهن- مکالی تعمیم یافته , خارج قسمت‌های خطی. Polymatroidal ideals, Regular decomposition function, Generic ideals, generalized Cohen-Macaulay ideals, Linear quotients. 65 75 http://mmr.khu.ac.ir/browse.php?a_code=A-10-1326-1&slc_lang=fa&sid=1 Somayeh Bandari سمیه بندری somayeh.bandari@yahoo.com 10031947532846005462 10031947532846005462 Yes Buein Zahra Technical University مرکز آموزش عالی فنی و مهندسی بوئین زهرا