  We illustrate the strength of Algebraic Methods, adapting Gaussian
Elimination and Substitution to the problem of Exact Boolean Satisfiability.
For 1-in-3 SAT with non-negated literals we are able to obtain considerably
smaller equivalent instances of 0/1 Integer Programming restricted to Equality
only.
  Both Gaussian Elimination and Substitution may be used in a processing step,
followed by a type of brute-force approach on the kernel thus obtained.
  Our method shows that Positive instances of 1-in-3 SAT may be reduced to
significantly smaller instances of I.P.E. in the following sense. Any such
instance of $|V|$ variables and $|C|$ clauses can be polynomial-time reduced to
an instance of 0/1 Integer Programming with Equality, of size at most $2/3|V|$
variables and at most $|C|$ clauses.
  We obtain an upper bound for the complexity of counting, $O(2\kappa r
2^{(1-\kappa) r})$ for number of variables $r$ and clauses to variables ratio
$\kappa$.
  We proceed to define formally the notion of a non-trivial kernel, defining
the problems considered as Constraint Satisfaction Problems.
  We conclude showing the methods presented here, giving a non-trivial kernel
for positive 1-in-3 SAT, imply the existence of a non-trivial kernel for 1-in-3
SAT.
