For a parametric algebraic system in finite fields, this paper presents a method for computing the cover and the refined cover based on the characteristic set method. From the cover, the author knows for what parametr...For a parametric algebraic system in finite fields, this paper presents a method for computing the cover and the refined cover based on the characteristic set method. From the cover, the author knows for what parametric values the system has solutions and at the same time presents the solutions in the form of proper chains. By the refined cover, the author gives a complete classification of the number of solutions for this system, that is, the author divides the parameter space into several disjoint components, and on every component the system has a fix number of solutions. Moreover, the author develops a method of quantifier elimination for first order formulas in finite fields.展开更多
This paper investigates the equality-constrained minimization of polynomial functions. Let R be the field of real numbers, and R[x1,..., xn] the ring of polynomials over R in variables x1,..., xn. For an f ∈ R[x1,......This paper investigates the equality-constrained minimization of polynomial functions. Let R be the field of real numbers, and R[x1,..., xn] the ring of polynomials over R in variables x1,..., xn. For an f ∈ R[x1,..., xn] and a finite subset H of R[x1,..., xn], denote by V(f : H) the set {f( ˉα) | ˉα∈ Rn, and h( ˉα) =0, ? h ∈ H}. We provide an effective algorithm for computing a finite set U of non-zero univariate polynomials such that the infimum inf V(f : H) of V(f : H) is a root of some polynomial in U whenever inf V(f : H) = ±∞.The strategies of this paper are decomposing a finite set of polynomials into triangular chains of polynomials and computing the so-called revised resultants. With the aid of the computer algebraic system Maple, our algorithm has been made into a general program to treat the equality-constrained minimization of polynomials with rational coefficients.展开更多
基金supported by the National 973 Program of China under Grant No.2011CB302400the National Natural Science Foundation of China under Grant No.60970152
文摘For a parametric algebraic system in finite fields, this paper presents a method for computing the cover and the refined cover based on the characteristic set method. From the cover, the author knows for what parametric values the system has solutions and at the same time presents the solutions in the form of proper chains. By the refined cover, the author gives a complete classification of the number of solutions for this system, that is, the author divides the parameter space into several disjoint components, and on every component the system has a fix number of solutions. Moreover, the author develops a method of quantifier elimination for first order formulas in finite fields.
基金supported by National Natural Science Foundation of China(Grant No.11161034)
文摘This paper investigates the equality-constrained minimization of polynomial functions. Let R be the field of real numbers, and R[x1,..., xn] the ring of polynomials over R in variables x1,..., xn. For an f ∈ R[x1,..., xn] and a finite subset H of R[x1,..., xn], denote by V(f : H) the set {f( ˉα) | ˉα∈ Rn, and h( ˉα) =0, ? h ∈ H}. We provide an effective algorithm for computing a finite set U of non-zero univariate polynomials such that the infimum inf V(f : H) of V(f : H) is a root of some polynomial in U whenever inf V(f : H) = ±∞.The strategies of this paper are decomposing a finite set of polynomials into triangular chains of polynomials and computing the so-called revised resultants. With the aid of the computer algebraic system Maple, our algorithm has been made into a general program to treat the equality-constrained minimization of polynomials with rational coefficients.
