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Wu's method of characteristic set
Wenjun Wu's method is an algorithm for solving multivariate polynomial equations introduced in the late 1970s by the Chinese mathematician Wen-Tsun Wu. This method is based on the mathematical concept of characteristic set introduced in the late 1940s by J.F. Ritt. It is fully independent of the Gröbner basis method, introduced by Bruno Buchberger (1965), even if Gröbner bases may be used to compute characteristic sets.
Wu's method is powerful for mechanical theorem proving in elementary geometry, and provides a complete decision process for certain classes of problem. It has been used in research in his laboratory (KLMM, Key Laboratory of Mathematics Mechanization in Chinese Academy of Science) and around the world. The main trends of research on Wu's method concern systems of polynomial equations of positive dimension and differential algebra where Ritt's results have been made effective. Wu's method has been applied in various scientific fields, like biology, computer vision, robot kinematics and especially automatic proofs in geometry.
Wu's method uses polynomial division to solve problems of the form:
where f is a polynomial equation and I is a conjunction of polynomial equations. The algorithm is complete for such problems over the complex domain.
The core idea of the algorithm is that you can divide one polynomial by another to give a remainder. Repeated division results in either the remainder vanishing (in which case the I implies f statement is true), or an irreducible remainder is left behind (in which case the statement is false).
More specifically, for an ideal I in the ring k[x1, ..., xn] over a field k, a (Ritt) characteristic set C of I is composed of a set of polynomials in I, which is in triangular shape: polynomials in C have distinct main variables (see the formal definition below). Given a characteristic set C of I, one can decide if a polynomial f is zero modulo I. That is, the membership test is checkable for I, provided a characteristic set of I.
A Ritt characteristic set is a finite set of polynomials in triangular form of an ideal. This triangular set satisfies certain minimal condition with respect to the Ritt ordering, and it preserves many interesting geometrical properties of the ideal. However it may not be its system of generators.
Let R be the multivariate polynomial ring k[x1, ..., xn] over a field k. The variables are ordered linearly according to their subscript: x1 < ... < xn. For a non-constant polynomial p in R, the greatest variable effectively presenting in p, called main variable or class, plays a particular role: p can be naturally regarded as a univariate polynomial in its main variable xk with coefficients in k[x1, ..., xk−1]. The degree of p as a univariate polynomial in its main variable is also called its main degree.
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Wu's method of characteristic set
Wenjun Wu's method is an algorithm for solving multivariate polynomial equations introduced in the late 1970s by the Chinese mathematician Wen-Tsun Wu. This method is based on the mathematical concept of characteristic set introduced in the late 1940s by J.F. Ritt. It is fully independent of the Gröbner basis method, introduced by Bruno Buchberger (1965), even if Gröbner bases may be used to compute characteristic sets.
Wu's method is powerful for mechanical theorem proving in elementary geometry, and provides a complete decision process for certain classes of problem. It has been used in research in his laboratory (KLMM, Key Laboratory of Mathematics Mechanization in Chinese Academy of Science) and around the world. The main trends of research on Wu's method concern systems of polynomial equations of positive dimension and differential algebra where Ritt's results have been made effective. Wu's method has been applied in various scientific fields, like biology, computer vision, robot kinematics and especially automatic proofs in geometry.
Wu's method uses polynomial division to solve problems of the form:
where f is a polynomial equation and I is a conjunction of polynomial equations. The algorithm is complete for such problems over the complex domain.
The core idea of the algorithm is that you can divide one polynomial by another to give a remainder. Repeated division results in either the remainder vanishing (in which case the I implies f statement is true), or an irreducible remainder is left behind (in which case the statement is false).
More specifically, for an ideal I in the ring k[x1, ..., xn] over a field k, a (Ritt) characteristic set C of I is composed of a set of polynomials in I, which is in triangular shape: polynomials in C have distinct main variables (see the formal definition below). Given a characteristic set C of I, one can decide if a polynomial f is zero modulo I. That is, the membership test is checkable for I, provided a characteristic set of I.
A Ritt characteristic set is a finite set of polynomials in triangular form of an ideal. This triangular set satisfies certain minimal condition with respect to the Ritt ordering, and it preserves many interesting geometrical properties of the ideal. However it may not be its system of generators.
Let R be the multivariate polynomial ring k[x1, ..., xn] over a field k. The variables are ordered linearly according to their subscript: x1 < ... < xn. For a non-constant polynomial p in R, the greatest variable effectively presenting in p, called main variable or class, plays a particular role: p can be naturally regarded as a univariate polynomial in its main variable xk with coefficients in k[x1, ..., xk−1]. The degree of p as a univariate polynomial in its main variable is also called its main degree.