By Sergey Abrahamyan (auth.), Vladimir P. Gerdt, Wolfram Koepf, Ernst W. Mayr, Evgenii V. Vorozhtsov (eds.)

The CASC Workshops are usually held in flip within the Commonwealth of IndependentStates(CIS)andoutsideCIS(Germanyinparticular,but,attimes, additionally different international locations with energetic CA communities). the former CASC Wo- store was once held in Japan, and the twelfth workshop used to be held for the ?rst time in Armenia, that is one of many CIS republics. it's going to be famous that greater than 35 institutes and scienti?c facilities functionality in the nationwide Academy of S- ences of Armenia (further information about the constitution of the academy could be foundhttp://www. sci. am). those associations are involved, specifically, with difficulties in such branches of ordinary technological know-how as arithmetic, informatics, physics, astronomy, biochemistry, and so on. It follows from the talks provided on the earlier CASC workshops that the tools and platforms of computing device algebra can be utilized effectively in the entire above-listed branches of normal sciences. accordingly, the organizers of the twelfth CASC Workshop desire that the current workshop might help the Armenian scientists to develop into much more accustomed to the services of complex desktop algebra equipment and structures and to get in contact with experts in desktop algebra from different international locations. The eleven prior CASC meetings, CASC 1998, CASC 1999, CASC 2000, CASC 2001, CASC 2002, CASC 2003, CASC 2004, CASC 2005, CASC 2006, CASC 2007, and CASC 2009 have been held, respectively, in St. Petersburg (R- sia), Munich (Germany), Samarkand (Uzbekistan), Konstanz (Germany), Yalta (Ukraine), Passau (Germany), St.

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**Extra info for Computer Algebra in Scientific Computing: 12th International Workshop, CASC 2010, Tsakhkadzor, Armenia, September 6-12, 2010. Proceedings**

**Sample text**

U(m) } be the set of diﬀerential indeterminates. The multivariate Δ-diﬀerential polynomial ring is given by F {U } := F {u(1) } . . {u(m) }. The (j) elements of U Δ := ui | i ∈ Zn≥0 , j ∈ {1, . . , m} are called diﬀerential variables. We remark, that the algebraic closure F of F is a diﬀerential ﬁeld with a differential structure uniquely deﬁned by the diﬀerential structure of F (cf. 2, Lemma 1]). Let m E := F [[z1 , . . , zn ]] ∼ =F U Δ j=1 with indeterminates z1 , . . , zn , where F [[z1 , .

2), which guarantees the continuation of solutions from lower ranking variables to higher ranking ones, also holds here. Any diﬀerential F -algebra R with a diﬀerential embedding of E → R might be chosen as universal set of solutions, for example a universal diﬀerential ﬁeld containing F : Clearly F [[z1 , . . , zn ]] embeds into its ﬁeld of quotients F ((z1 , . . , zn )), and thus F [[z1 , . . , zn ]] also embeds into a universal diﬀerential ﬁeld containing F , since F ((z1 , . . , zn )) is a ﬁnitely generated diﬀerential ﬁeld extension of F (cf.

A pseudo reduction procedure and several splitting algorithms on the basis of polynomial remainder sequences are introduced as tools for the main algorithm, which is presented at the end of the section. 1 Preliminaries Let F be a computable ﬁeld of characteristic 0 and R := F [x1 , . . , xn ] the polynomial ring in n variables. A total order < on the indeterminates of R is called a ranking. The notation R = F [x1 , . . , xn ] shall implicitly deﬁne the ranking x1 < . . < xn . The indeterminate x is called leader of p ∈ R if x is the <-largest variable occurring in p and we write ld(p) = x.