  The new type of ideal basis introduced herein constitutes a compromise
between the Gr\"obner bases based on the Buchberger's algorithm and the
characteristic sets based on the Wu's method. It reduces the complexity of the
traditional Gr\"obner bases and subdues the notorious intermediate expression
swell problem and intermediate coefficient swell problem to a substantial
extent. The computation of an $S$-polynomial for the new bases requires at most
$O(m\ln^2m\ln\ln m)$ word operations whereas $O(m^6\ln^2m)$ word operations are
requisite in the Buchberger's algorithm. Here $m$ denotes the upper bound for
the numbers of terms both in the leading coefficients and for the rest of the
polynomials. The new bases are for zero-dimensional polynomial ideals and based
on univariate pseudo-divisions. However in contrast to the pseudo-divisions in
the Wu's method for the characteristic sets, the new bases retain the algebraic
information of the original ideal and in particular, solve the ideal membership
problem. In order to determine the authentic factors of the eliminant, we
analyze the multipliers of the pseudo-divisions and develop an algorithm over
principal quotient rings with zero divisors.
