Category:ApCoCoA-1:Package ncpoly

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Package ncpoly is designed to enable us to do basic calculations with polynomials in non-commutative polynomial rings (or free associative algebras), over the field of rational numbers Q or over finite fields Z/(p) where p is a prime. For instance, in the package, there are functions for addition of two (non-commutative) polynomials (via Add(F1,F2)), subtraction (via Sub(F1,F2)), multiplication (via Mul(F1,F2)), getting the leading word (via Lw(F)) and leading coefficient (via Lc(F)) of a non-zero polynomial, computing the normal remainder (via NR(F,G)) of a polynomial w.r.t a LIST of polynomials, interreducing (via Interreduction(G)) a LIST of polynomials, etc. Moreover, the package also contains functions for Groebner basis computations. For example, there are functions to check if a LIST of polynomials is a Groebner basis (via IsGB(G)), enumerate (reduced) (partial) Groebner bases (via GB(G[,Optimize,OFlag,DB,LB]) and RedGB(G[,Optimize,OFlag,DB,LB])), and compute truncated Groebner bases (via TruncatedGB(G[,Optimize,OFlag,DB])) of finitely generated (two-sided) ideals. Consequently, we can apply the package to many algebraic applications, i.e. to compute Macaulay basis (via MB(Gb)) and the values of Hilbert function (via HF(Gb)), compute generating systems for leading term ideals, kernel of K-algebra homomorphism, intersection of ideals, etc.

Important issues about this package:

(a) Predefined alias for this package is as follows.

             Alias NC := $apcocoa/ncpoly;

(b) The very first step to use functions in this package is to set non-commutative polynomial ring environment via the command


For instance, the following command

             Use QQ[x[1..2],y[1..2]];

sets the ring to be the non-commutative polynomial ring generated by {x[1],x[2],y[1],y[2]} over the rational numbers. Note that, for the time being, the package only supports non-commutative polynomial rings over the rational field QQ and finite fields ZZ/(p) where p is a prime.

(c) The word ordering is set via the function


where the parameter Ordering is a STRING indicating which ordering we are working with. Note that word orderings are induced by the order of indeterminates. And, for the time being, the package supports the following word orderings, and among which "LLEX" is the default ordering.

       (b1) "LLEX": the length-lexicographic ordering

       (b2) "ELIM": an elimination ordering

       (b3) "LRLEX": the length-reverse-lexicographic ordering

       (b4) "DEGREVLEX": the degree-reverse-lexicographic ordering

For instance, the following commands

             Use ZZ/(2)[a,b,c,d];


define the non-commutative polynomial ring generated by {a,b,c,d} over the binary field {0,1}, and set the word ordering to be the length-lexicographic ordering induced by a>b>c>d.

(d) One can use the function


to get basic information on the working polynomial ring.

Representation of non-commutative polynomials in this package