Let \(R_ 0\) be a domain with the field of fractions \(K_ 0\) and let \(S_ 0\) be a multiplicative set of \(R_ 0\). For a natural number \(n\), let \(R_ n=R_ 0[X_ 1,\ldots,X_ n]\) and \(P_ n=K_ 0[X_ 1,\ldots,X_ n]\). An algebraic system is a finite set \(E=\{e_ 1,\ldots,e_ k\}\) of \(R_ n\). To each prime ideal \(I\) in \(P_ n\), let \(I_ i=I\cap R_ i\), \(A_ i=R_ i/I_ i\) and \(K_ i\) the field of fractions of \(A_ i\). Then the set of fields \(K_ 0,\ldots,K_ n\) is called the tower associated with the prime ideal \(I\). If \(f\in R_ i\backslash R_{i-1}\), the main variable of \(f\) is defined to be \(X_ i\) and its index to be \(i\). Further, the degree of \(f\) means its degree in \(X_ i\) and the leading coefficient is its coefficient in \(R_{i-1}\) of the highest power of \(X_ i\). The corresponding functions are denoted respectively by main(\(f\)), index(\(f\)), \(\deg(f)\), lc(\(f\)). Using this terminology, the author defines:
Definition 3.2. A triangular set in \(R_ n\) is a list of non-constant polynomials \((f_ 1,\ldots,f_ k)\) in \(R_ n\) satisfying the following for \(i=1,\ldots,k\):
(1) [weak triangular] \(\text{index}(f_ j)<\text{index}(f_ i)\) for \(j<i\); then inductively, let \(K_ j=K_{j-1}[X_ j]/(f_ i)\) if \(\text{index}(f_ i)=j\) and \(qf(K_{j-1}[X_ j])\) (the total quotient ring) otherwise:
(2) [reduced] the degree of \(f_ i\) in main\((f_ j)\) is strictly less than \(\deg(f_ j)\) for \(j=1,\ldots,i-1\);
(3) [normalized] \(\text{index(lc}^ h(f_ i))\notin \{\text{index}(f_ 1),\ldots,\text{index}(f_{i-1})\}\) for any \(h>0\);
(4) [\(R_ 0\)-normalized] if \(\text{lc}^ h(f_ i))\in R_ 0\), then \(\text{lc}^ h(f_ i))\in S_ 0\);
(5) [square-free] the resultant of \(f_ i\) and its derivative with respect to \(\text{main}(f_ i)\) is invertible in \(K_{j-1}\), where \(j=\text{index}(f_ i)\);
(6) [primitive] the coefficients of \(f_ i\) viewed as a multivariate polynomial over \(K_{j-1}[X_ j]\) generate the unit ideal of \(K_{j- 1}[X_ j]\) for all \(j<\text{index}(f_ i)\).
And, the author remarks:
Proposition 3.3. Let \(I\) be a prime ideal, and \(K_ 0,\ldots,K_ n\) the associated tower; if \(K_ i\) is algebraic over \(K_{i-1}\), the minimal polynomial of \(x_ i\) over \(K_{i-1}\) may be chosen in \(R_ i\) such that the set of these polynomials forms a triangular set. There is exactly one such choice.
Proposition 3.4. Let \(f_ 1,\ldots,f_ k\) be a triangular set, \(h=\text{lc}(f_ 1)\cdots\text{lc}(f_ k)\) and let \(I=\{p\in P_ n\mid h^ ep\in(f_ 1,\ldots,f_ k)\}\). Then if the triangular system is associated with a prime ideal \(J\), we have \(I=J\).
Corollary 3.5. The set of generators of the ideal \(I\) above is computable from the triangular set.
Then the author shows how to compute numerically the common zeros of a given prime ideal, using the associated triangular set.