PolynomialsandGröbnerBases · ETHZürich$–StudentSeminar$in$Combinatorics:$Mathemacal$So

ETH Zürich – Student Seminar in Combinatorics: Mathema:cal So<ware Polynomials and Gröbner Bases -‐ Alice Feldmann – 16.12.2014

Polynomials and Gröbner Bases

Alice Feldmann 16th December 2014

ETH Zürich Student Seminar in Combinatorics:

Mathema:cal So<ware Prof. K. Fukuda

Autumn Semester 2014


Talk Outline

I.  Introduc:on: Computa:onal Algebra II.  Polynomials and Gröbner Bases

a.  Algebraic Background b.  Gröbner Bases c.  Buchberger‘s Algorithm

III.  Mathema:cal So<ware for Problems in Computa:onal Algebra a.  Singular b.  CoCoA c.  Gfan

IV.  Implementa:on: Buchberger‘s Algorithm in Singular V.  References


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.   Gröbner Bases ... 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)

Introduc:on: Computa:onal Algebra Polynomials and Gröbner Bases Mathema:cal So<ware for Problems in Computa:onal Algebra Implementa:on: Buchberger‘s Algorithm in Singular


Polynomials and Gröbner Bases Algebraic Background

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.  Term: product of nonzero element c of a field F and a monomial. IV.  Polynomial p in n variables: sum of a finite number of terms with coefficients in F

for


Algebraic Background Gröbner Bases Buchberger‘s Algorithm


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




Example

Consider the polynomials:

Observe:

Conclude:




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?




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




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: .




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: .




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.






Gröbner Bases

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.




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.




Buchberger‘s Algorithm

Mo9va9on Takes an arbitrary genera:ng set for an ideal I and produces a Gröbner Basis G for I from it.

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:

(Observe: )

Apply Buchberger‘s Criterion: A finite set is a Gröbner Basis of I if and only if for all pairs .




Buchberger‘s Algorithm

Input: Output: a Gröbner Basis for with

REPEAT

FOR each pair in G‘ DO

IF THEN UNTIL




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.


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


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


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


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


Singular CoCoA Gfan


Implementa9on: Buchberger‘s Algorithm 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



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)

Algebraic Background: David A. Cox, John Liile, Donal O‘Shea: Graduate Texts in Mathema:cs -‐ Using Algebraic Geometry (2005) Ralf Fröberg: An Introduc:on to Gröbner Bases (1997)

SoRware: hip://www.singular.uni-‐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)

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