(* ---------------------------------------------------------------------------- * $Id: RRINT.md,v 1.1 1994/03/11 15:21:50 pesch Exp $ * ---------------------------------------------------------------------------- * This file is part of MAS. * ---------------------------------------------------------------------------- * Copyright (c) 1993 Universitaet Passau * ---------------------------------------------------------------------------- * $Log: RRINT.md,v $ * Revision 1.1 1994/03/11 15:21:50 pesch * Counting real roots of multivariate polynomials, Diplomarbeit F. Lippold * * ---------------------------------------------------------------------------- *) DEFINITION MODULE RRINT; (* Real Root Integral Definition Module *) (* Import lists and declarations *) FROM MASSTOR IMPORT LIST;CONSTrcsid = "$Id: RRINT.md,v 1.1 1994/03/11 15:21:50 pesch Exp $";CONSTcopyright = "Copyright (c) 1993 Universitaet Passau";PROCEDURE RRIPIQ(c: LIST;VARA,a: LIST); (* Real root integral polynomial integral quotient. A is an integral polynomial. a and c are nonzero integers. New values for A and a are computed such that the equation A/a := (1/c)*(A/a) holds. If a and the content of A have gcd 1 then this is also true for the result. *)PROCEDURE RRIPQSUM(B,b,c: LIST;VARA,a: LIST); (* Real root integral polynomial quotient sum. A and B are integral polynomials. a,b and c are non zero integers. New values for A and a are computed such that the equation A/a := A/a + c*(B/b) holds. If b and the content of B have gcd 1 and a and the content of A have gcd 1 then the new integral polynomial A has gcd 1 with the new value of a. *)PROCEDURE RRINFORM(G,R: LIST;VARa,NF: LIST); (* Real root integral normal form. G reduced integral groebner basis of a nontrivial zerodimensional ideal, R is the set of reduced terms. NF is a list of entries (u,ut,ua,up) with: u is an element of R * R, RRREDTEST(G,u,_,ut), up is an integral polynomial and ua is an integer such that up/ua is the normal form of u wrt G. a is the lcm of all integers ua in NF. The elements of NF are sorted with respect to the actual termorder in decreasing order of the first entry. *)PROCEDURE RRISTRCONST(G,R: LIST;VARU,a,beta: LIST); (* Real root integral structure constants. G reduced integral groebner basis of a nontrivial zerodimensional ideal, R the set of reduced terms. beta is the integral matrix of combined structure constants beta[u,v] wrt the basis a*R with u from U = R * R and v from R. U and the rows of the matrix beta are sorted with respect to the actual termorder in increasing order. *)PROCEDURE RRIVARMATRICES(G,R,U,beta: LIST;VARal,L: LIST); (* Real root integral multiplication matrices of variables. G reduced integral groebner basis of a nontrivial zerodimensional ideal. R is the set of reduced terms in increasing order, U= R * R and beta is the set of combined structure constants wrt c*R for some nonzero integer c. al=(a(1),...,a(n)) is a list of integers and L is a list of entries of the form (1,M(i)) and M(i) is the matrix of multiplication with a(i)*X(i) wrt c*R. *)PROCEDURE RRIPOLMATRIX(R,h,f,fl: LIST;VARMh,L: LIST); (* Real root integral polynomial matrix. R is the set of reduced terms in increasing order, h is a polynomial of domain D, f is a positive integer, fl=(f(1),...,f(n)) is a list of positive integers, L contains nonempty lists L(i) of the form j(1),M(1),j(2),M(2),...,j(k),M(k) with 1=j(1)<j(2)<...<j(k) and M(l) is the integral matrix of multiplikation with (f*f(i)*X(i))**j(l) for the variable X(i). L is extended with new calculated matrices. If d(i) is the maximal degree in the variable X(i) and d is the maximal total degree of the terms of the polynomial h, then Mh is the integral matrix of multiplication with (f**d)*(f(1)**d(1))*...*(f(1)**d(1)) * h. *)PROCEDURE RRIQUADFORM(R,U,beta,Mh: LIST): LIST; (* Real root integral quadratic form. R is the set of reduced terms in increasing order, U = R * R and beta is the set of integral combined structure constants wrt a*R for some nonzero integer a. Mh is the integral matrix of multiplication with c*h for some positive constant c represented columnwise. The matrix Q = (q(i,j)) with q(i,j) = tr(M(h)*M(v(i))*M(v(j)) and v(i),v(j) from a*R is computed. *)PROCEDURE RRICOUNT(G,H,v,tf: LIST): LIST; (* Real root integral count. G is an integral reduced groebner basis of a zerodimensional ideal I, H is a list of polynomials of length s. v is a vector of signs with length not greater than s. tf is the trace-flag. ZNL is a list of pairs (z,n) with n is an element of {-1,0,+1}**s and z > 0 is the number of real zeroes of I wrt the sign condition n for the elements of H. ZNL is sorted wrt the invers lexicographical order of the n. If there does not exist any real zero or a zero satisfiing the sign condition v, then the empty list is returned. *)ENDRRINT. (* -EOF- *)