The eight files mentioned at the end of this page give all S
polynomials needed for the paper Groebner
basis for the 2X2 determinantal ideal mod t^2 (hereafter
refered to as the paper). The
following instructions may help decipher the data files:
Each file contains all possible pairs of S polynomials
S(\alpha,\beta), \alpha from the first family, \beta from the second
family.
All notation is as in the paper.
The data for a particular S(\alpha,\beta) is demarcated by
----------------------------------
The first pieces of data for a particular S polynomial
S(\alpha,\beta) are simply the polynomials \alpha and
\beta.
In the cases where the S polynomials are not relatively prime,
the following is the sequence of data written to the file after \alpha
and \beta:
Lead term of the S-polynomial.
The first divisor \gamma (from the family \mathcal H).
The quotient f_{\gamma}.
The lead term of the product f_{\gamma} \gamma (to be compared
against the lead term of the S polynomial.
..and so on, to the last divisor, the last quotient, and the
lead term of the last product f_{\gamma} \gamma (so that in the end,
S(\alpha,\beta)
should equal sum_{\gamma} f_{\gamma} \gamma).
A confirmation that the difference S(\alpha,\beta) -
sum_{\gamma} f_{\gamma} \gamma is zero.
A confirmation that for each gamma, lead (f_{\gamma} \gamma)
was less than lead(S-polynomial).
The formula for the S-polynomial
Rewrite: this is the expanded form of sum_{\gamma}
f_{\gamma} \gamma
The final equality S(\alpha,\beta) = sum_{\gamma} f_{\gamma}
\gamma in a form suitable for export to TeX: Here, the nonstandard
macros \del, \eps and \lam are simply short forms of \delta, \epsilon,
and \lambda.