Border bases for lattice ideals

详细信息    查看全文

• 作者：Giandomenico Boffi^a ; ¹ ; ^{giandomenico.boffi@unint.eu" class="auth_mail" title="E-mail the corresponding author} ; Alessandro Logar^b ; ² ; ^{logar@units.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Border basis ; Grö ; bner basis ; Lattice ideal ; Maximal clique ; Maximum clique ; Order ideal
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：March-April 2017
• 年：2017
• 卷：79
• 期：part_P1
• 页码：43-56
• 全文大小：435 K

文摘

The main ingredient to construct an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300840&_mathId=si1.gif&_user=111111111&_pii=S0747717116300840&_rdoc=1&_issn=07477171&md5=538f6e82ccbbf0ed732267c7858c6ee9" title="Click to view the MathML source">Oclass="mathContainer hidden">class="mathCode">$\mathcal{O}$-border basis of an ideal class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300840&_mathId=si2.gif&_user=111111111&_pii=S0747717116300840&_rdoc=1&_issn=07477171&md5=785122d86e352731543e0fb53e1af4cf" title="Click to view the MathML source">I⊆K[x₁,…,x_n]class="mathContainer hidden">class="mathCode">$I\subseteq K[x_1,\dots ,x_n]$ is the order ideal class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300840&_mathId=si1.gif&_user=111111111&_pii=S0747717116300840&_rdoc=1&_issn=07477171&md5=538f6e82ccbbf0ed732267c7858c6ee9" title="Click to view the MathML source">Oclass="mathContainer hidden">class="mathCode">$\mathcal{O}$, which is a basis of the K  -vector space class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300840&_mathId=si3.gif&_user=111111111&_pii=S0747717116300840&_rdoc=1&_issn=07477171&md5=5b26a01fc1fa074b180fe6a8d4a1fc37" title="Click to view the MathML source">K[x₁,…,x_n]/Iclass="mathContainer hidden">class="mathCode">$K[x_1,\dots ,x_n]/I$. In this paper we give a procedure to find all the possible order ideals associated with a lattice ideal class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300840&_mathId=si328.gif&_user=111111111&_pii=S0747717116300840&_rdoc=1&_issn=07477171&md5=446b69f7e3c704fe12ebda2f43557f76" title="Click to view the MathML source">I_Mclass="mathContainer hidden">class="mathCode">$I_M$ (where M   is a lattice of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300840&_mathId=si488.gif&_user=111111111&_pii=S0747717116300840&_rdoc=1&_issn=07477171&md5=eedeaefc69b3f4726548d57724062924" title="Click to view the MathML source">Zⁿclass="mathContainer hidden">class="mathCode">${\mathbb{Z}}^n$). The construction can be applied to ideals of any dimension (not only zero-dimensional) and shows that the possible order ideals are always in a finite number. For lattice ideals of positive dimension we also show that, although a border basis is infinite, it can be defined in finite terms. Furthermore we give an example which proves that not all border bases of a lattice ideal come from Gröbner bases. Finally, we give a complete and explicit description of all the border bases for ideals class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300840&_mathId=si328.gif&_user=111111111&_pii=S0747717116300840&_rdoc=1&_issn=07477171&md5=446b69f7e3c704fe12ebda2f43557f76" title="Click to view the MathML source">I_Mclass="mathContainer hidden">class="mathCode">$I_M$ in case M   is a 2-dimensional lattice contained in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300840&_mathId=si128.gif&_user=111111111&_pii=S0747717116300840&_rdoc=1&_issn=07477171&md5=9334e81614693006a31649fae144c10b" title="Click to view the MathML source">Z²class="mathContainer hidden">class="mathCode">${\mathbb{Z}}^2$.