LEADER 04293nam 22005535 450 001 9910437867603321 005 20220415193347.0 010 $a4-431-54574-3 024 7 $a10.1007/978-4-431-54574-3 035 $a(CKB)3710000000083794 035 $a(EBL)1636790 035 $a(OCoLC)902410365 035 $a(SSID)ssj0001129752 035 $a(PQKBManifestationID)11742625 035 $a(PQKBTitleCode)TC0001129752 035 $a(PQKBWorkID)11079261 035 $a(PQKB)11064936 035 $a(DE-He213)978-4-431-54574-3 035 $a(MiAaPQ)EBC1636790 035 $a(PPN)176126341 035 $a(EXLCZ)993710000000083794 100 $a20140107d2013 u| 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aGröbner bases $estatistics and software systems /$fedited by Takayuki Hibi 205 $a1st ed. 2013. 210 1$aTokyo :$cSpringer Japan :$cImprint: Springer,$d2013. 215 $a1 online resource (488 p.) 300 $aIncludes index. 311 $a4-431-54573-5 327 $aA Quick Introduction to Gröbner Bases -- Warm-up Drills and Tips for Mathematical Software -- Computation of Gröbner Bases -- Markov Bases and Designed Experiments -- Convex Polytopes and Gröbner Bases -- Gröbner Basis for Rings of Differential Operators and Applications -- Examples and Exercises.  . 330 $aThe idea of the Gröbner basis first appeared in a 1927 paper by F. S. Macaulay, who succeeded in creating a combinatorial characterization of the Hilbert functions of homogeneous ideals of the polynomial ring. Later, the modern definition of the Gröbner basis was independently introduced by Heisuke Hironaka in 1964 and Bruno Buchberger in 1965. However, after the discovery of the notion of the Gröbner basis by Hironaka and Buchberger, it was not actively pursued for 20 years. A breakthrough was made in the mid-1980s by David Bayer and Michael Stillman, who created the Macaulay computer algebra system with the help of the Gröbner basis. Since then, rapid development on the Gröbner basis has been achieved by many researchers, including Bernd Sturmfels. This book serves as a standard bible of the Gröbner basis, for which the harmony of theory, application, and computation are indispensable. It provides all the fundamentals for graduate students to learn the ABC?s of the Gröbner basis, requiring no special knowledge to understand those basic points. Starting from the introductory performance of the Gröbner basis (Chapter 1), a trip around mathematical software follows (Chapter 2). Then comes a deep discussion of how to compute the Gröbner basis (Chapter 3). These three chapters may be regarded as the first act of a mathematical play. The second act opens with topics on algebraic statistics (Chapter 4), a fascinating research area where the Gröbner basis of a toric ideal is a fundamental tool of the Markov chain Monte Carlo method. Moreover, the Gröbner basis of a toric ideal has had a great influence on the study of convex polytopes (Chapter 5). In addition, the Gröbner basis of the ring of differential operators gives effective algorithms on holonomic functions (Chapter 6). The third act (Chapter 7) is a collection of concrete examples and problems for Chapters 4, 5 and 6 emphasizing computation by using various software systems. 606 $aStatistics  606 $aStatistical Theory and Methods$3https://scigraph.springernature.com/ontologies/product-market-codes/S11001 606 $aStatistics and Computing/Statistics Programs$3https://scigraph.springernature.com/ontologies/product-market-codes/S12008 606 $aStatistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences$3https://scigraph.springernature.com/ontologies/product-market-codes/S17020 615 0$aStatistics . 615 14$aStatistical Theory and Methods. 615 24$aStatistics and Computing/Statistics Programs. 615 24$aStatistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences. 676 $a005.55 702 $aHibi$b Takayuki$4edt$4http://id.loc.gov/vocabulary/relators/edt 906 $aBOOK 912 $a9910437867603321 996 $aGröbner Bases$92539314 997 $aUNINA