RUR Of Solutions To Polynomial Systems And Its Application

Polynomial system solving is a classic topic with a long history,which is still full of vitality today.It plays a very important role in the theoretical research of various areas of mathematics and practical applications in the field of engineering.Polynomial systems can be divided into zero-dimensional and positive-dimensional,according to the number of solutions to them on an algebraic closure of the base field.The zero-dimensional systems have a finite number of solutions while the positive-dimensional ones have an uncountable infinite number of solutions.It also means that each solution to a zero-dimensional system is isolated and the solutions to a positive-dimensional one must contain at least a solution manifold.Rational Univariate Representation(RUR)is a symbolic method for zero-dimensional system solving.It writes each coordinate component of the solutions to a zero-dimensional system as a rational function of the solutions to a single univariate polynomial.So this means that RUR provides an explicit expression or a parametric form of the solutions to a zero-dimensional system and makes it possible to perform more easily arithmetic op-erations to solution coordinates,such as addition,subtraction,multiplication,division,and even interpolation etc.Hence,attentions are gradually being paid to it.Separating elements are at the heart of RUR theory.A separating element is a polynomial(usually a linear one)satisfying that it gets different values when evaluated at different zeros of the system.Specifically,let k be a field of characteristic 0,I(?)k[x1,...,xn]be a zero-dimensional ideal,then the variety Vk(I)of I is a finite point set,and a separating element t ? k[x1,...,xn]of I satisfies that ?,? ? Vk(I)with ??? implies t(?)? t(?).Once a separating element t is determined,the RUR of I associated to t can be computed and is a(n + 2)tuple of univariate polynomials in k[T],i.e.{Xt(T),Gt(1,T),Gt(xi,T),i=1,...,n?,(5)where Xt(T)is the characteristic polynomial of the multiplication matrix of t in the quo-tient ring k[x1,...,xn]/I.The zeros of Xt(T)are exactly t{?),?? Vk(I),and the multi-plicity of t(a)as a zero of Xt(T)is the same as a as a zero of I.From(5),the variety of I can be expressed asVk(I)={(Gt(x1,T0)/Gt(1,T0),...,Gt)(xn,T0)/Gt(1,T0)T0?Vk(Xt(T)},(6)i.e.the zeros of I can be described by those of Xt(T).Hence,after a RUR of I is obtained,solving the equation Xt(T)= 0 and substituting the zeros into the coordinate expressions in(6)produce all the zeros of I.In other words,from a RUR of I,performing univariate polynomial equation solving once and carrying out univariate rational function evaluating n times export all the zeros of I.For the zeros of a positive-dimensional polynomial ideal I,to get a similar result to(6),Tan Chang and Zhang Shugong address the notion of Rational Representation Set(RRS)in 2010,express Vk(I)with a finite of RRS's,and finally propose the theory of Rational Representation for positive-dimensional systems.The definition of Rational Representation Set is rephrased as following.Definition 1(Rational Representation Set)Let I(?)k[X]be a positive-dimensional ideal.Let U(?)X,#U = d,1 ? d<n,and V:= X\U = ?v1,...,vn-d}.Suppose U is maximally algebraically independent modulo I,t ? k[V]is a separating element of Ie,and{Xu,t(T),GU,t(1,T),GU,t(v1,T),...,Gu,t(vn-d,T)} is the RUR of Ie associated to t.Then we call RtU:= {F(U)?t(U),XU,t(T),GU,t(1,T),GU,t(v1,T),...,GU,t,(vn-d,T)}(7)a Rational Representation Set(RRS)of I associated to U and t,where ?t(U)? k[U]de-notes the numerator of ResT(XU,t(T),(?)XU,t(T)/(?)T),with XU,t(T)the squarefree part of XU,t(T).Actually,A RRS is a RUR of the ideal Ie(base field is the rational function field k(U)),and can be considered as introducing parameters U into(5)to represent part of zeros of I.For U0 ? kd,F(U0)? 0 guarantees that coefficients(rational functions in(?) k(U))of polynomials in(7)make sense and can be evaluated at U0.For UO ? k with F(U0)?0,?t(U0)? 0 ensures that we can get zeros of I using(7)which satisfy U=U0.This is to say that,in general,only part of zeros of I with F(U)?t(U)? 0 can be obtained from a RRS.To express all the zeros of I,by using the idea of Wu's method,we need to compute a RRS of(I,F(U)?t(U)).But the RRS also may represent only part of zeros of<I,F((U)?t(U)>.So we should repeat the process.By ACC,the process terminates in finite steps.In other words,we can get a finite number of Rational Representation Set Rj,j = 1,...,s,to represent the zeros of I.The conclusion is stated as following.Theorem 1 Let I(?)k[X]be a positive-dimensional ideal.Then Vk(I)=U s j=1 Wj,s?N,(?) where Wj can be represented by a Rational Representation Set Rj.U j=1 s {Rj} is called a Rational Representation(RR)of Vk(I).For convenience,we also call U j=1 s {Rj} a RR of I.This is the Rational Representation theory of positive-dimensional ideals.The RUR theory of zero-dimensional ideals and the RR theory of positive-dimensional ideals are collectively referred to as the RUR theory of polynomial ideals.In this paper,we mainly study this theory and its applications in theory research and engineering.The main results of the paper are as follows:1.Propose the Simplified Rational Representation for positive-dimensional ideals.Remove some RRS's from the RR of a positive-dimensional ideal I,so represent the variety Vk(I)with less RRS's,then simplify and speed up the computing.After the RRS of I associated to an indeterminate set U and a separating element t(notice that it is a separating element of Ie)is obtained asRtU:= {F(U)?t(U),XU,t(T),FU,t(1,T),GU,t(v1,T),...,GU,t(Vn-d,T)},the RRS of<I,F(U)· ?t(U)>needs to be computed,and it is equivalent to computing the RRS's of<I,F(U)>and<I,?t(U)>respectively.But in practical,because ?t(U)is the numerator of the resultant of two univariate polynomials with rational functions their coefficients,it is usually with a high degree,so leads to a hard computation of(I,?t(U)).When the RRS RtU of I is obtained,the variety of I can be represented asVk(I)=WtU U Vk(<I,F(U)>)?(Vk(<I,?t(U)>)\Vk(<I,F(U)>)),where WtU is the zeros of I can be gained from RtU,and it represents the zeros of I satisfying F(U)· ?t(U)? 0.Note that,intuitively Vk((I,?t(U)>)\Vk(<I,F(U)))can be considered as the intersection of some components of Vk(I),so due to the closed property of a variety,they can be approached by some zeros in Vk(I).Using the structure of vector space of k(U)[V]/Ie and the continuity of eigenvalues of a matrix w.r.t its elements,we prove that the points in Vk(<I,?t(U)>)\Vk((I,F(U)>)can be approached by some points in WtU,and it means that RtU can "represent" VtU:= WtU?(Vk(I,?t)\Vk(I,F)).Here,the meaning of "represent" includes taking limits of a sequence of points in WtU.Hence,the variety of I can be represented as Vk(I)= VtU?Vk(<I,F(U)>).Now,we can define the Simplified Rational Representation Set(SRRS)of I associated to U and t asSRtU:={F(U),XU,t(T),GU,t(1,T),GUt(v1,T),...,GU,t(vn-d,T)?,and it "represents" the zero set VtU.It can be considered to represent more zeros of I,since in the meaning of set theory VtU is "bigger" than WtU.Similar to the RR the-ory,we prove that a finite number of SRRS' s can represent the variety of a positive-dimensional ideal,and propose the Simplified Rational Representation(SRR)theory for positive-dimensional ideals.From the above discussion,an SRR of I doesn't contain the RRS of(I,?t(U)>,which may involve operations of polynomials with high degrees,so the computing is simplified.2.Propose the notion of U-prime ideals,by using which the characteristic of ideals satisfying I = Iec can be algebraically explained more clear;discuss the properties of U-prime ideals with dimension 1,and prove that their SRR's can be formed by at most two SRRS's.Given U c X,a proper ideal I(?)k[X]is called U-prime,if 0 ? f · g ? I,f ? k[U],g ?k[X]implies g ?I.We prove that,if U(?)X is maximally algebraically independent modulo I,then I = Iec iff I is U-prime.Also,using the properties of U-prime ideals,we prove that I is U-prime iff every minimal primary component of i is U-prime.It enables an algebraic explanation of the intuitive conclusion that if i = Iec,then every minimal primary component J of I satisfies J = Jec.If dim(I)= 1,i.e.?U = 1,the characteristic of the reduced Grobner basis of I w.r.t a block order<(U<<V)helps us simplify the computation of F(U)in the RRS of I;if I is U-prime with dim(I)= 1,we prove that a nonzero polynomial f(U)? k[U]makes<I,f(U)>a zero-dimensional ideal or(1),so its SRR can be formed by at most two SRRS's.3.For a zero-dimensional ideal with a Shape basis,based on eigenvalue method,propose a method to compute its RUR from the transformational matrix of two bases of the quotient ring of the ideal as a vector space.Suppose that I is a zero-dimensional ideal with a Shape basis.Let B1:= {?D,...,?1 =1} be a monomial quotient ring basis of k[X]/I w.r.t a monomial order<,where ?i-1<?i,i = 2,...,D,with D being the dimension of k[X]/I as a k-vector space.Then the ideal I must have a separating element t satisfying that B2:= {ti,i = 0,...,D-1} is a basis of the quotient ring k[X]/I.Denote TB1,B2 the transformation matrix from B1 to B2,then B2 ? TB1,B2 B1 mod I.Next,we show how to compute a RUR of I using TB1,B2.Compute Xt(T).Since B2 is a basis of k[X]/I,the minimal polynomial of t as an operator on k[X]/I is the same as Xt(T).So solving the system of linear equations(?)[CO,...,CD-1]TB1,B2 = tDleads to Xt(T)= TD-?i=0 D-1 CiTi,where tD is the(row)vector of tD w.r.t B1.Compute Gt(xi,T),i = 1,...,n.Note that in fact the RUR theory is to represent indeterminates with rational functions of a separating element,and the transformational matrix TB1,B2 from B1 to B2 is expressing ti as linear combinations of elements of B1,i.e.B2 ? TB1,B2B1 mod I.(8)So the inverse transformation of TB1,B2 gives an expression of indeterminates with the pow-ers of t,i.e.B1 ? TB1,B2-1 B2 mod I.Then we can denote Xi ? Pi`(t)mod I,deg(Pi'(T))<D,i = 1,...,n.Note that Pi' is of a high degree,so to get a PUR of the ideal,we denote Pi(T):= Rem(Pi(T),Xt(T)),i=1,...,n.Because Xt(T)?(?),{Xt(T),P1(T),...,Pn(T)}is the PUR of I associated to t.Given Gt(l,T),to compute a RUR of I,write Gt(xi,T)= Rem(Pi(T)·Gt(1,T),Xt(T)),i = 1,...,n,then {Xt(T),Gt(1,T),Gt(x1,T),...,Gt(xn,T)} is the RUR of I associated to t.4.Discuss the application of the RUR theory of zero-dimensional ideals in algebraic geometry.Let {Xt(T),Gt(1,T),Gt(xi,T),i= 1,...,n} be the RUR of a zero-dimensional ideal I associated to a separating element t,and Xt(T)= f1?1(T)· · · fs?s(T)be the factorization of Xt(T)in k[T].We prove:a.Gt(1,T)is a constant in k iff I has exactly a zero.b.The radical of I is<fi(t)· · · fs(t),Gt(1,t)x1-Gt(x1,t),...,Gt(1,t)xn-Gt(xn,t)>.c.The prime decomposition of(?)is ?i=1 s<fi(t),Gt(1,t)x1-Gt(x1,t),...,Gt(1,t)xn-Gt(xn,t).5.Discuss the application of Simplified Rational Representation in the SHEPWM problem.Improve the process to transform the trigonometric SHEPWM equations into poly-nomial ones.Based on a new formula xi =(-1)i+1 2 cos ?i and the explicit transformation formula between elementary symmetric polynomials and power sum symmetric polyno-mials,we give the explicit formulation of the SHEPWM equations.Compared with others,we get equations with shorter coefficients,and gain a great speedup.For example,if the number of switching angles N = 6,the transformation is finished in 123.281s with others'method implemented in Mathematica 11,while our code in Maple 2016 only uses 0.125s.The computing of SRR is performed in a rational function field.It leads to the prob-lem of Intermediate Expression Well.So we consider to use Homomorphism method to speed up.First,we get a series of zero-dimensional ideals via evaluating the modulation index in SHEPWM equations at a series of points.The we compute their RUR's using the RUR theory.Last,SRR of the SHEPWM problem can be lifted from these RUR's.The advantage of "SRR" with the modulation index m a parameter is improving the efficiency of computing the switching angles when the value of m changes.When N = 5,after computing switching angles for 460 modulation indices,we get the average time 0.0284s for one modulation index.