Square-free pure triangular decomposition of zero-dimensional polynomial systems (Q6594999)

Triangular sets are finite lists of non-constant polynomials arranged in a specific order based on their leading variables, where each polynomial's leading variable is less than that of the subsequent polynomials. They play a crucial role in computational algebra, particularly in solving polynomial systems. A triangular set \(T = [T_1, \ldots , T_t]\) is said to be square-free, if the discriminant of \(T_1\) with respect to its leading variable is not zero, and for each \(i\) the resultant of the separant of \(T_i\) and \([T_1, \ldots , T_i]\) for any \(i\) is zero. Triangular decomposition is a process that transforms a system of polynomials into a finite set of triangular sets.\N\NThe paper presents the concept of Square-Free Pure Triangular Decomposition (SFPTD) for zero-dimensional polynomial systems and introduces an algorithm for its computation using Gröbner bases. It is proven that the arithmetic complexity of this algorithm can be single exponential in the square of the number of variables. Furthermore, this approach allows for the computation of isolating cubes of real solutions to the system and the radical of the ideal generated by the system. The algorithm has been implemented in Maple 2021, and its efficiency is compared to classical methods. The authors conclude that SFPTD could be a key approach for addressing challenges in computational algebraic geometry.