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View File Name : cassou.m2

export {"cassou"}

cassou = method();
cassou(Ring) := kk -> (
    (b,c,d,e) := (symbol b, symbol c, symbol d, symbol e);
    R := kk[b,c,d,e];
    { 15*b^4*c*d^2 + 6*b^4*c^3 + 21*b^4*c^2*d - 144*b^2*c - 8*b^2*c^2*e - 28*b^2*c*d*e - 648*b^2*d + 36*b^2*d^2*e + 9*b^4*d^3 - 120,
      30*c^3*b^4*d - 32*d*e^2*c - 720*d*b^2*c - 24*c^3*b^2*e - 432*c^2*b^2 + 576*e*c - 576*d*e + 16*c*b^2*d^2*e + 16*d^2*e^2 + 16*e^2*c^2 + 9*c^4*b^4 + 5184  + 39*d^2*b^4*c^2 + 18*d^3*b^4*c - 432*d^2*b^2  + 24*d^3*b^2*e - 16*c^2*b^2*d*e - 240*c,
      216*d*b^2*c - 162*d^2*b^2 - 81*c^2*b^2 + 5184 + 1008*e*c - 1008*d*e + 15*c^2*b^2*d*e - 15*c^3*b^2*e - 80*d*e^2*c + 40*d^2*e^2  + 40*e^2*c^2,
      261 + 4*d*b^2*c - 3*d^2*b^2 - 4*c^2*b^2 + 22*e*c - 22*d*e	
       }
)

beginDocumentation()

doc /// 
    Key
    	cassou
	(cassou,Ring)
    Headline
    	the system of Pierrette Cassou-Nogu`es
    Usage
    	cassou(kk)
    Inputs
    	kk:Ring
	    the coefficient ring
    Outputs
    	:List	
	    of the polynomials in the system
    Description
    	Text
	   This system was solved in May 2020, using @TO solveSystem@ in Macaulay2 v1.15
	     with an Intel(R) Core(TM) i5-4258U CPU at 2.40GHz
	   
	    There were 2 solutions found in 39.0405 seconds (with a Bezout bound of 1344).
	    
	    Reference: 
	    Obtained by electronic mail by Carlo Traverso.
	    See the POSSO test suite.

            T.Y. Li, Tianjun Wang, Xiaoshen Wang: 
	    "Random Product Homotopy with Minimal BKK Bound",
	    in "The Mathematics of Numerical Analysis",
	    Edited by Renegar, J. and Shub, M. and Smale, S.,
	    Lectures in Applied Mathematics vol 32, 1996.
	    Proceedings of the AMS-SIAM Summer Seminar in Applied Mathematics,
	    Park City, Utah, July 17-August 11, 1995, Park City, Utah".
	    
	    See also: http://homepages.math.uic.edu/~jan/Demo/cassou.html
	Example
	    cassou(RR_53)
    ///