Polynomials and Gröbner Bases

Transcription

1 Alice Feldmann 16th December 2014 ETH Zürich Student Seminar in Combinatorics: Mathema:cal So<ware Prof. K. Fukuda Autumn Semester 2014

2 Talk Outline I. Introduc:on: Computa:onal Algebra II. a. b. c. III. a. Singular b. CoCoA c. Gfan IV. V. References

3 Introduc9on: Computa9onal Algebra Tackling problems with algebraic methods using the computer... For example... I. Solving inhomogeneous Systems of Linear Equa9ons Everybody knows the Gauss algorithm for solving systems of linear equa:ons! Use matrices as algebraic tool and an appropriate so<ware (for example MATLAB) to analyse whether there exists a unique solu:on for an inhomogeneous system (which happens if and only if det(m) 0). II.... provide solu:ons to basic problems in ring theory, for example: Let R be a ring and I be an ideal in R. Given r R, is r I? Membership Problem... provide an approach for solving systems of non- linear equa:ons (reduc:on of number of variables)

4 Let s remember some basic algebraic defini:ons and concepts... I. Monomial over a collec:on of variables II. Total degree of a monomial III. IV. Term: product of nonzero element c of a field F and a monomial. Polynomial p in n variables: sum of a finite number of terms with coefficients in F for

5 V. Polynomial Ring: set of all polynomials in n variables with coefficients in F VI. Field of Ra9onal Func9ons: VII. Ideal: non- empty subset of field F, such that a. for and b. For for an arbitrary polynomial VIII. Ideal generated by polynomials Remark: is the smallest ideal, which contains

6 Example Consider the polynomials: Observe: Conclude:

7 Remarks I. Hilbert Basis Theorem Every ideal I in has a finite genera:ng set. This means, for any given ideal I, there exists a finite set of polynomials such that. II. Division Algorithm in Given two polynomials, we can divide f by g, producing a unique quo:ent q and remainder r such that and it is either r = 0, or r has degree strictly smaller than the degree of g. If there exists a division algorithm for polynomials in just one variable, is there one for polynomials in many variables?

8 Monomial Orders Formal defini9on: A monomial order on is any rela:on > on the set of monomials in sa:sfying: I. > is a total (linear) ordering rela:on. II. > is compa4ble with mul4plica4on in. III. > is a well- ordering, i.e. every non- empty collec:on of monomials has a smallest element under >. Examples: I. Lexicographic Order II. Graded Reverse Lexicographic Order

9 Examples I. Lexicographic Order Let and be monomials in. We say, if in the difference the le<- most nonzero entry is posi:ve. Example I: For, we have, because the le<- most entry of the difference vector must be posi:ve:. Example II: We have, because. II. Graded Reverse Lexicographic Order Let and be monomials in. We say, if either, or in case the sums are equal, the right- most entry is nega:ve in the difference. Example I: We have, because the of the total degrees. Example II: We have, because the right- most entry of the difference vector must be nega:ve:.

10 Leading Term of a Polynomial Let be a polynomial in. Fix any monomial order >. The leading term of f with respect to > is the product, where is the largest monomial appearing in in the ordering >. Denote the leading term by. Example: Consider. For the lexicographic order, we have:. However, for the Graded Reverse Lexicographic Order, we have:.

11 Division Algorithm in Fix any monomial order > in and let be an ordered s- tuple of polynomials in. Then every can be wriien as where, and either r = 0, or r is a linear combina:on of monomials, of which none is divisible by any of. Remarks: I. r is called the remainder of f on division by F and is some:mes denoted by. II. The division algorithm allows to divide f by an s- tuple of polynomials. III. The outcome of the division depends on the choice of the monomial order.

12 Mo9va9on By applying the division algorithm, one can decide whether a given is a member of a given ideal. If the remainder of f on division by F is zero, then we have, and by defini:on. Recall on of our previous examples, where we defined. We have already showed, that but if we divide p by using our division algorithm, we obtain a nonzero remainder:. The reason is that none of the leading terms of or divides the leading term of p.,

13 Defini9on Fix a monomial order > on and let be an ideal. A Gröbner Basis for I (with respect to >) is a finite collec:on of polynomials with the property that for every nonzero polynomial is divisible by for some i. A Gröbner Basis G for an ideal I is called reduced, if for all dis:nct, no monomial appearing in is a mul:ple of. monic, if it is reduced and if the leading coefficient of all polynomials is 1. Remarks I. A Gröbner Basis G is indeed a basis for I, i.e. II. Let G be a Gröbner Basis for I and then the remainder of f divided by G is zero, i.e. III. The remainder obtained through division by polynomials of a Gröbner Basis for an ideal is unique.

14 Mo9va9on Takes an arbitrary genera:ng set Basis G for I from it. for an ideal I and produces a Gröbner How does it work? Let be nonzero polynomials. Fix a monomial order and let and, where Furthermore, define as the least common mul:ple of the leading terms. Now define the S- polynomial of f and g as: Apply Buchberger s Criterion: A finite set is a Gröbner Basis of I if and only if for all pairs. (Observe: )

15 Input: Output: a Gröbner Basis for with REPEAT FOR each pair in G DO UNTIL IF THEN

16 Example Consider the polynomials and. Fix the monomial ordering to be the lexicographic ordering. We have that, which yields Let s divide the S- polynomial by : we obtain the remainder Note that the leading term is not divisible by the leading terms of f and g. Hence, adjoin to our set of polynomials F and consider the two new S- polynomials and. Con:nue with this procedure and adjoin the remainders of further divisions if there are nonzero...

17 Singular CoCoA Gfan Mathema9cal SoRware for Computa9onal Algebra Introduc9on there exist a lot of possibili:es for execu:ng computa:onal algebra see list on wikipedia: hip://en.wikipedia.org/wiki/list_of_computer_algebra_systems (not complete...) Examples: Singular CoCoA Gfan

18 Singular CoCoA Gfan SINGULAR computer algebra system for polynomial computa:ons emphasis on commuta4ve algebra, algebraic geometry and singularity theory started in 1984 (Berlin), now: seiled at the University of Kaiserslautern free and open- source so<ware under the GNU General Public Licence main objects: ideals and modules in polynomial rings over fields or quo:ent rings features: computa:on of Gröbner bases (Buchberger s and Mora s Algorithm), polynomial factoriza:on, resultants, characteris:c sets and gcd opera:ons, classifica:on of singulari:es wriien in C

19 Singular CoCoA Gfan CoCoA Computa:ons in Commuta:ve Algebra emphasis on computa:ons in mul4variate polynomial rings over ra:onal integers and on their ideals and modules ini:ated in 1987 to perform calcula:ons using Buchberger s algorithm (Genova) free so<ware features CoCoALib (open- source C++ GPL library) libraries available for applica:ons in various areas (sta:s:cs, integer programming,...) kernel is wriien in C, user writes in Pascal- like syntax features efficient calcula:ons with very big integers and ra:onal numbers and implementa:ons of Buchberger s algorithm using different orderings

20 Singular CoCoA Gfan Gfan so<ware package for compu:ng Gröbner fans of polynomial ideals in and tropical varie4es based on Buchberger s algorithm emphasis on tropical and polyhedral geometry using algebraic methods started in 2003, wriien by A. N. Jensen (University of Aarhus, Denmark), supported by Prof. Fukuda wriien in C++ high performance (calcula:ng 3000 reduced Gröbner bases in 12s) various subprograms enable fast computa:ons and easy handling

21 Implementa9on: in Singular Programming Basics language very similar to C/C++ (all inputs must be terminated by ;, types for variables, loops, func:ons etc.) non- trivial algorithms always require the defini:on of a ring at the beginning! ring syntax: ring name = (coefficients), (name of ring variables), (ordering) ; implemented algorithms for opera:ons on ideals, polynomials, etc. for example: factoriza:on of polynomials into irreducible factors (Cantor- Zassenhaus Algorithm), compu:ng the determinant of a square matrix, calcula:ng the greatest common divisor of polynomials most interes:ng feature: computa:on of Gröbner bases standard features like characteriza:on- func:on (typeof), type conversion,... include libraries with LIB library name ; various orderings: grevlex ordering dp, lex ordering lp, block ordering possibility of execu:ng the division algorithm, returns quo:ent and remainder

22 References Computa9onal Algebra K. G. Fischer, P. Loustaunau, J. Shapiro, E. L. Green, D. Farkas: Lecture Notes in Pure and Applied Mathema:cs Computa:onal Algebra (1994) : David A. Cox, John Liile, Donal O Shea: Graduate Texts in Mathema:cs - Using Algebraic Geometry (2005) Ralf Fröberg: An Introduc:on to (1997) SoRware: hip:// kl.de hip://cocoa.dima.unige.it/flyer4.html hip://home.math.au.dk/jensen/so<ware/gfan/gfan.html hip://en.wikipedia.org/wiki/list_of_computer_algebra_systems Implementa9on: Groebner Bases Sta:s:cs and So<ware Systems. Takayuki Hibi, Springer Japan, 2013 Singular Manual (online)