| 1 | Degree functions for the leading term (wrt. the monomial ordering) |
|---|
| 2 | ================================================================== |
|---|
| 3 | |
|---|
| 4 | p_Deg: a degree that is compatible with the monomial ordering |
|---|
| 5 | (i.e. for dp: totaldegree, for lp(lex): exponent of first variable(s), etc.) |
|---|
| 6 | p_FDeg: a degree for use in GB computations: |
|---|
| 7 | (i.e. p_Deg for degree orderings, p_Totaldegree for block orderings (lex) |
|---|
| 8 | or mixed orderings). |
|---|
| 9 | Requirement: there are only finitely many monomials of the same degree |
|---|
| 10 | Remark: A weight 0 (or positive and negative weights) allow infinitely |
|---|
| 11 | many monomials of the same degree which lead to infinite cycles in GB |
|---|
| 12 | computations - p_FDeg uses p_Totaldegree in this case |
|---|
| 13 | p_Totaldegree: the sum of all exponents (the usual total degree) |
|---|
| 14 | p_WTotaldegree: the weighted sum of all exponents(the usual weigthed degree)*OrdSgn |
|---|
| 15 | For matrix orderings this is the scalar product of the first row |
|---|
| 16 | of the ordering matrix with the exponent vector |
|---|
| 17 | p_WDegree: the weighted sum of all exponents (obsolete, currently not used): |
|---|
| 18 | Difference to p_WTotaldegree: |
|---|
| 19 | - does not muliply by OrdSgn |
|---|
| 20 | - uses p_WFirstTotalDegree for the first block |
|---|
| 21 | - for all but the first block: use the first non-zero entry in each |
|---|
| 22 | column of the ordering matrix as weight (and not the entry in the |
|---|
| 23 | first row) |
|---|
| 24 | p_WFirstTotalDegree: the weighted sum of the exponents of the first block |
|---|
| 25 | (scalar product of the first row of the ordering matrix |
|---|
| 26 | with the exponent vector) |
|---|
| 27 | |
|---|
| 28 | Degree function for a polynomial |
|---|
| 29 | ================================== |
|---|
| 30 | p_LDeg: return the length and the maximal degree wrt. p_FDeg |
|---|
| 31 | |
|---|
| 32 | Ring properties wrt. monomial orderings |
|---|
| 33 | ======================================= |
|---|
| 34 | OrdSgn: 1 for well orderings (i.e. 1 is the smallest monomial) |
|---|
| 35 | -1 otherwise |
|---|
| 36 | MixedOrder: 0 for well orderings |
|---|
| 37 | and local orderings (i.e. 1 is the largest monomial) |
|---|
| 38 | 1 otherwise |
|---|
| 39 | LexOrder: 1 for block orderings (incl. lex), 0 otherwise |
|---|
| 40 | |
|---|
