% Copyright 2015 by Jonathan D. Hauenstein

% Deflate family from Example 5.2 
% N. Li and L. Zhi
% Computing the multiplicity structure of an isolated singular solution: case of breadth one
% Journal of Symbolic Computation, 47, 700-710, 2012.

function [Fd,Vd] = LiZhi52_deflate(s)
    % Input: s - integer >= 2
    % Output: Fd - deflated polynomial system
    %         Vd - variables of deflated polynomial system

    % error checking
    Fd = []; Vd = [];
    if nargin < 1 || s < 2
        disp(['The degree d must be at least 2!']);
        return;
    end;

    % setup the original system
    X = sym('x',[1,s]);
    F = [X(end)^3];
    for j = s-1:-1:1
        F = [X(j)^2 + X(j) - X(j+1); F];
    end;
    Vd = X;
    
    % we need to perform 2 deflations using the upper left hand block
    % first deflation -- adds one new equation
    J = jacobian(F,X);
    A = J(1:end-1,1:end-1);
    B = J(1:end-1,end);    
    dA = det(A);
    AB = A\B;
    N = dA*[-AB;1];
    F1 = J(end,:)*N;

    % second deflation -- uses same upper left hand block 
    % this adds another new equation
    J1 = jacobian(F1,X);
    Fd = [F;F1;J1*N];

    % display summary of deflation
    disp(['Deflated system']);
    disp(['  functions: ',num2str(length(Fd))]);
    disp(['  variables: ',num2str(length(Vd))]);
return;