توصيفگر ها :
حلقه چندجمله اي ها , پايه گربنر , الگوريتم بوخبرگر , ايده آل همگن , حلقه موضعي , پايه استاندارد , تابع هيلبرت , چندگانگي , فضاي دوگان , الگوريتم ديتون زنگ
چكيده فارسي :
كي از موضوعات مهم در ھندسه جبري حل دستگاهھاي معادلات چندجملهاي است كه كاربردھاي فراواني در رياضيات
و علوم مهندسي دارد. از طرفي يكي از ابزارھاي نمادين براي حل چنين دستگاهھايي پايهھاي گربنر ھستند كه با استفاده
از آنھا ميتوان براي يك دستگاه چندجملهاي داده شده، يك دستگاه چندجملهاي جديدي كه معادل آن است معرفي كرد
كه داراي خاصيت مثلثي نسبت به متغيرھا باشد. اما يكي از نكات مهم براي دستگاهھاي معادلات چندجملهاي كه كاربرد
فراواني در تحليل دستگاهھاي ديناميكي دارد، بررسي چندگانگي يك ريشه يك دستگاه داده شده است. در طي سالها
روشهاي گوناگوني براي اين منظور ارائه شده است كه استفاده از روشهاي جبري و بخصوص پايه استاندارد، همواره در
اولويت بوده است. محاسبه ساختارهاي چندگانگي به طور گسترده در هندسه جبري مورد مطالعه قرار گرفته است. يكي
ديگر از روشهاي مرسوم براي محاسبه چندگانگي استفاده از فضاي دوگان است. در اين پاياننامه ضمن معرفي مقدمات
مورد نياز در زمينه فضاي دوگان به كاربرد آن )با استفاده از ماتريس مكالي( براي محاسبه چندگانگي خواهيم پرداخت.
چكيده انگليسي :
A Gröbner basis for a polynomial ideal provides interesting computational information about the ideal. For example,
we can list the dimension of the ideal, its Hilbert function, a way to answer the ideal membership question, and so
on. Computing a Gröbner basis is a well understood problem at least in the setting of exact computation, for example
using the Buchberger algorithm. Roughly the same can be said about ideals in a local polynomial ring. In the exact
setting we can compute a standard basis (the local equivalent of a Gröbner basis) using variations of Buchberger’s
algorithm such as those using Mora’s normal form algorithm.
However in many practical situations, using only exact computations becomes infeasible and we are forced to
rely on approximate numerical data. For example many large systems of polynomials can only be solved in practice
with numerical algorithms such as homotopy continuation. We may want to investigate the properties of a given
ideal in the local ring at some solution point, but this point is only known to us approximately. Although we can
approximate the point with arbitrarily high precision, the error can never be entirely eliminated. In this context the
usual algorithms for computing a standard basis are unsuitable because they are not numerically stable. In numerical
setting, if we find some information about the ideal, we shall check the properties carefully to be sure about them.
In this thesis we will focus on an approach which uses purely local information: computing the local dual space
of the ideal. The dual space is the vector space of all functionals that annihilate every element of the ideal. This
idea was first developed in the seminal work of Macaulay. The dual space of an ideal can provide much of the same
information as a standard basis, such as the Hilbert function of the ideal, and a test for ideal membership.
There are several algorithms for computing the dual space of an ideal in a local ring. In this direction, we are
interested in the truncated computation of standard bases. One that will be discussed in this thesis is the Dayton–Zeng
algorithm and another is the Mourrain algorithm, although both are based on the ideas of Macaulay. The numerical
advantage to dual space algorithms is that they reduce the problem to finding the kernel of a matrix. This can be done
in a numerically stable way using singular value decomposition (SVD) method.
These truncated dual space algorithms provide a way to fully characterize the local properties of an ideal when
the ideal is zero-dimensional, i.e. the point of interest is an isolated solution. In this case the dual space has finite
dimension, so truncating at a high enough dimension we will find a basis for the whole space. However, when the
ideal is not zero-dimensional or when the dimension is not known at priori, this strategy will be studied as well. This
M.Sc. thesis is based on the following papers
• Dayton, Barry H and Zeng, Zhonggang., Computing the multiplicity structure in solving polynomial systems,
international symposium on symbolic and algebraic computation ISSAC’05, Beijing, China, 116-123 (2005).
• Krone, Robert., Numerical algorithms for dual bases of positive-dimensional ideals, Jornal of Algebra and
Application, VOL(12), 1350018, (2013)
