In this paper, we study the relationship between iterated resultant and multivariate discriminant. We show that, for generic form f（xn） with even degree d, if the polynomial is squarefreed after each iteration, the ...In this paper, we study the relationship between iterated resultant and multivariate discriminant. We show that, for generic form f（xn） with even degree d, if the polynomial is squarefreed after each iteration, the multivariate discriminant A（f） is a factor of the squarefreed iterated resulrant. In fact, we find a factor Hp（f, [x1 , xn]） of the squarefreed iterated resultant, and prove that the multivariate discriminant A（f） is a factor of Hp（f,[x1,... ,xn]）. Moreover, we conjecture that Hp（f, [x1,..., xn]） =△（f） holds for generic form f, and show that it is true for generic trivariate form f（x, y, z）.展开更多
In this article it is proved that there exist a large number of polynomials which have small discriminant in terms of the Euclidean and p-adic metrics simultaneously. The measure of the set of points which satisfy cer...In this article it is proved that there exist a large number of polynomials which have small discriminant in terms of the Euclidean and p-adic metrics simultaneously. The measure of the set of points which satisfy certain polynomial and derivative conditions is also determined.展开更多
We define the second discriminant D_(2)of a univariate polynomial f of degree greater than 2 as the product of the linear forms 2r_(k)-r_(i)-r_(j)for all triples of roots r_(i),r_(k),r_(j)of f with i<j and j≠k,k≠...We define the second discriminant D_(2)of a univariate polynomial f of degree greater than 2 as the product of the linear forms 2r_(k)-r_(i)-r_(j)for all triples of roots r_(i),r_(k),r_(j)of f with i<j and j≠k,k≠i.D_(2)vanishes if and only if f has at least one root which is equal to the average of two other roots.We show that D_(2)can be expressed as the resultant of f and a determinant formed with the derivatives of f,establishing a new relation between the roots and the coefficients of f.We prove several notable properties and present an application of D_(2).展开更多
The theory of primitive polynomials over Galois rings is analogue to the same one over finite fields. It also provides useful tools for one to study the maximal period sequences over Galois rings. In the case of F<...The theory of primitive polynomials over Galois rings is analogue to the same one over finite fields. It also provides useful tools for one to study the maximal period sequences over Galois rings. In the case of F<sub>q</sub>, we have more complete results. In the case of Z<sub>p<sup>n</sup></sub>, n≥2, there are also some results. In particular, according to refs. [3, 4] and using the technique of trace representation of maximal period sequences over F<sub>q</sub>, we have found a discriminant which can judge whether a given polynomial f（x） over Z<sub>p<sup>n</sup></sub> is a primitive polynomial if f（x） mod p is a primitive polynomial over F<sub>p</sub>. Furthermore, it is easy to calculate the discriminant using the coefficients of f（x）.展开更多
The compound KdV-type equation with nonlinear terms of any order is reduced to the integral form. Using the complete discrimination system for polynomial, its all possible exact traveling wave solutions are obtained. ...The compound KdV-type equation with nonlinear terms of any order is reduced to the integral form. Using the complete discrimination system for polynomial, its all possible exact traveling wave solutions are obtained. Among those, a lot of solutions are new.展开更多
A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral fo...A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.展开更多
An extended Fan's algebraic method is used for constructing exact traveling wave solution of nonlinearpartial differential equations.The key idea of this method is to introduce an auxiliary ordinary differential e...An extended Fan's algebraic method is used for constructing exact traveling wave solution of nonlinearpartial differential equations.The key idea of this method is to introduce an auxiliary ordinary differential equationwhich is regarded as an extended elliptic equation and whose degree Υ is expanded to the case of r>4.The efficiency ofthe method is demonstrated by the KdV equation and the variant Boussinesq equations.The results indicate that themethod not only offers all solutions obtained by using Fu's and Fan's methods,but also some new solutions.展开更多
文摘In this paper, we study the relationship between iterated resultant and multivariate discriminant. We show that, for generic form f（xn） with even degree d, if the polynomial is squarefreed after each iteration, the multivariate discriminant A（f） is a factor of the squarefreed iterated resulrant. In fact, we find a factor Hp（f, [x1 , xn]） of the squarefreed iterated resultant, and prove that the multivariate discriminant A（f） is a factor of Hp（f,[x1,... ,xn]）. Moreover, we conjecture that Hp（f, [x1,..., xn]） =△（f） holds for generic form f, and show that it is true for generic trivariate form f（x, y, z）.
基金supported by the Science Foundation Ireland Programme (Grant No. RFP/MTH1512)
文摘In this article it is proved that there exist a large number of polynomials which have small discriminant in terms of the Euclidean and p-adic metrics simultaneously. The measure of the set of points which satisfy certain polynomial and derivative conditions is also determined.
基金supported by National Natural Science Foundation of China(Grant Nos.61702025 and 11801101)the Special Fund for Guangxi Bagui Scholar Project+1 种基金Guangxi Science and Technology Program(Grant No.2017AD23056)the Startup Foundation for Advanced Talents in Guangxi University for Nationalities(Grant No.2015MDQD018)。
文摘We define the second discriminant D_(2)of a univariate polynomial f of degree greater than 2 as the product of the linear forms 2r_(k)-r_(i)-r_(j)for all triples of roots r_(i),r_(k),r_(j)of f with i<j and j≠k,k≠i.D_(2)vanishes if and only if f has at least one root which is equal to the average of two other roots.We show that D_(2)can be expressed as the resultant of f and a determinant formed with the derivatives of f,establishing a new relation between the roots and the coefficients of f.We prove several notable properties and present an application of D_(2).
文摘The theory of primitive polynomials over Galois rings is analogue to the same one over finite fields. It also provides useful tools for one to study the maximal period sequences over Galois rings. In the case of F<sub>q</sub>, we have more complete results. In the case of Z<sub>p<sup>n</sup></sub>, n≥2, there are also some results. In particular, according to refs. [3, 4] and using the technique of trace representation of maximal period sequences over F<sub>q</sub>, we have found a discriminant which can judge whether a given polynomial f（x） over Z<sub>p<sup>n</sup></sub> is a primitive polynomial if f（x） mod p is a primitive polynomial over F<sub>p</sub>. Furthermore, it is easy to calculate the discriminant using the coefficients of f（x）.
基金The project supported by Scientific Reseaxch Fund of Education Department of Heilongjiang Province of China under Grant No. 11511008
文摘The compound KdV-type equation with nonlinear terms of any order is reduced to the integral form. Using the complete discrimination system for polynomial, its all possible exact traveling wave solutions are obtained. Among those, a lot of solutions are new.
文摘A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.
基金National Natural Science Foundation of China under Grant No.10672053
文摘An extended Fan's algebraic method is used for constructing exact traveling wave solution of nonlinearpartial differential equations.The key idea of this method is to introduce an auxiliary ordinary differential equationwhich is regarded as an extended elliptic equation and whose degree Υ is expanded to the case of r>4.The efficiency ofthe method is demonstrated by the KdV equation and the variant Boussinesq equations.The results indicate that themethod not only offers all solutions obtained by using Fu's and Fan's methods,but also some new solutions.
