Groebner bases over noetherian path algebras Universite Cheikh Anta Diop Colloque sur la cryptographie et les codes correcteurs Dakar 3-11 decembre 2015 Andre Saint Eudes Mialebama Bouesso Postdoc fellow Stellenbosch University African Institute for Mathematical Sciences South Africa mialebama@aims.ac.za December 8, 2015
Overview What is a Groebner basis? Path algebras - path orders Computing a Groebner basis over noetherian path algebras
What is a Groebner basis? What is a Groebner basis? Let K be a field and I = hf 1,...,f s i be a finitely generated ideal of K[x 1,...,x n ]. By a Groebner basis of I we mean a generating set of I satisfying to an additional condition. Condition Let f 2 K[x 1,...,x n ] \{0} be a nonzero polynomial, then by the division algorithm there exist polynomials q 1,...,q s, h 2 K[x 1,...,x n ]suchthatf = q 1 f 1 +...+ q s f s + h where h is not necessarily unique. A generating set of I for which h is unique is called Groebner basis. Remark The following algorithm computes a Groebner basis for an ideal.
Algorithm Buchberger s algorithm Input : g 1,...,g s 2 R and a monomial order <. Output : Groebner basis G for I = hg 1,...,g s i initialization: G := {g 1,...,g s } repeat G := G 0 for each pair g i, g j 2 G 0 do S(g i, g j ) 1appleiapplejapples! G 0 + S if S 6= 0then G := G 0 [ {S} end end until G = G 0 ;
Path algebras Definitions By a directed graph or a quiver we mean a quadruple =( 0, 1, r, s) where 0 is the set of vertices, 1 the set of edges and r, s : 1! 0 be maps. If e 2 1 is an edge, then s(e) isthe source of e and r(e) is the range of e. A sequence of edges = e 1...e n such that r(e i )=s(e i+1 )for i =1,...,n 1iscalledpathin. In this case we denote s( ) =s(e 1 )andr( ) =r(e n ). If is a path, then the number of edges in denoted L( ) iscalled the length of. A vertex is regarded as a path of length zero. A closed path is a path such that s( ) =r( ). A cycle is closed path which doesn t admit a loop at any of its vertices. If and are paths, then we define the multiplication as follow: is the path adjoining and by concatenation if r( ) =s( ); otherwise we get zero. By a path algebra K we mean the set of all linear combinations of paths in with coe cients in K.
Paths order Let be the set of all paths in. By a monomial we mean a path and by a polynomial we mean a linear combination of paths. Definition A well-ordering is a total ordering with the condition that every subset of the set of paths has a least element. A well-ordering < is called admissible if it satisfies the following conditions: for all p, q, r, s 2, 1. p < q =) rps < rqs whenever rps and rqs are both nonzero; 2. q = spr and q 6= p =) q > p.
Paths order Definition Let p = e 1 e r and q = f 1 f l be two paths in E, thenwesay that p is less than q with respect to the left-lexicographic order and we denote p < lex q if there exists a path m such that p = me k e r, q = me s f l and e k < f s. Remark The left lexicographic order is not necessarily a well-ordering thus not necessarily a left-admissible ordering. Definition Let p = e 1 e r and q = f 1 f l be two paths in E, thenwesay that p is less than q with respect to the length left-lexicographic order and we denote p < llex q if l(p) < l(q) orl(p) =l(q) and p < lex q.
Division with remainder, determinate version Theorem Let > be an admissible order and the path algebra K,and f 1,...,f s 2 K \{0}. Foreveryg 2 K,thereexistsauniquely sx sx sx determined expression g = p i f i q i + m i f i + f i r i + h i=1 (where for all i; p i, q i, m i, r i and h are elements of K )satisfying to: DD1 For i > j, no term of p i Lt(f i )q i, m i Lt(f i )andlt(f i )r i is divisible by Lt(f i ). i=1 DD2 For all i, no term of h is divisible by Lt(f i ). i=1
Noetherian path algebra Definition A graph is called noetherian if its corresponding path algebra is noetherian. Proposition A graph containing two non-identical cycles that intersect at a vertex is not noetherian. Proposition A graph is not noetherian if and only if, if contains a cycle C, an edge not occurring in C, with its source in C, and an edge not occurring in C, with its range in C.
Groebner bases over noetherian path algebras S-polynomial Let f 1,...,f r 2 K be nonzero polynomials and < be an admissible order. For each pair of indicies i, j, the S-polynomial S(f i, f j ) is defined as follow: 8 < m ji f i f j m ij if A hold S(f i, f j )= f : i m ji m ij f j if B hold 0 otherwise. Where m ji = lcm(lt(f i), Lt(f j )) and Lt(f i ) A- r(lm(f j )) = s(lm(f i )) or the last edge in Lm(f j )isequalto the first edge in Lm(f i ); B- r(lm(f i )) = s(lm(f j )) or the last edge in Lm(f i )isequalto the first edge in Lm(f j );
Groebner bases over noetherian path algebras Computing Note that S(f i, f j )= S(f j, f i ) 8 i, j, itactuallysu ces to consider the S(f i, f j )withj < i. Wecanevendobetter:fori =2, 3,...,r, consider the monomial ideal M i = hlt(f 1 ),...,Lt(f i 1 )i : hlt(f i )i = h lcm(lt(f i), Lt(f j )) /j =1,...,i Lt(f i ) 1i = hm ji /j < ii. As it turn out, there no need to consider all the m ji with j < i (i,e there is no need to consider all the corresponding S-polynomial S(f i, f j )) in our test, for each i, and for each minimal monomial generator x of M i, consider exactly one j = j(i, ) < i such that m ji involves x, and compute a standard expression for S(f i, f j ) with remainder h i,.
Groebner bases over noetherian path algebras Theorem: Buchberger s criterion Let < be an admissible order and G = {f 1,...,f r } K.With notation as above, G is a Groebner basis if and only if all remainders h i, are zero. Theorem: Buchberger s algorithm Given I = hf 1,...,f r i 1. Set k = r; 2. For each i =2, 3,...,k and for each minimal monomial generator x of M i, compute a remainder h i, as described above; 3. If some h i, is nonzero, set k = k +1, f k = h i,, and go back to step 2; 4. Return f 1,...,f k.
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