First examples on computing resultants in Maplef := x^2-2;
g := x^2+1; with(LinearAlgebra):S := SylvesterMatrix(f,g,x);Determinant(S);resultant(f,g,x);A second example in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGPQ== where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSgmIzg0NzQ7RicvRjBGPUY5LUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieUYnRi9GMkY5RjlGOQ==f := x*y-1;
g := x^2+y^2-4;S := SylvesterMatrix(f,g,x);Determinant(S);R := resultant(f,g,x);To apply Proposition 9 of 3.5 to find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiQkYnRi9GMi1GNjYtUSgmbm90aW47RidGOUY7L0Y/Rj1GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlgtRiw2JVEoJiM4NDg0O0YnL0YwRj1GOS1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUYsNiVRInlGJ0YvRjJGOUY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUZpbjYmLUYjNiQtRiw2JVEieEYnRi9GMkY5RjlGYG9GY29GOQ== such that 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 where 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 we first solve 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 for 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 (using the extended Euclidean algorithm) then multiply through by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to clear fractions.gcdex(f,g,x,'s','t');s;t;A := simplify(R*s);
B := simplify(R*t);A*f+B*g=R;simplify(A*f+B*g-R);We had this rational parametrization of the circle: 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 , 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 . To find the implicit equation f(x,y) = 0 we eliminate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== from the polynomials 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 and 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 .t := 't'; # clear assignment to tf := (1+t^2)*x-(1-t^2);
g := (1+t^2)*y-2*t;S := SylvesterMatrix(f,g,t);Determinant(S);resultant(f,g,t);Maple uses several algorithms to compute the resultant. It does not expand the determinant of Sylvesters matrix. It turns out that one can modify the Euclidean algorithm to compute the resultant and this is the basis for an efficient algorithm. The basis for this algorithm is covered in exercises 14,15,16,&17 in section 3.5. Maple uses a modification to this algorithm which avoids computing with fractions. I will present the details of the algorithm in the MACM 401 Introduction to Computer Algebra course in Spring 2015.TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzOTQyOTEyMjI1ODc4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjMSIlIiUiIiIiIiFGJkYnRidGJkYnRiYhIiNGJ0YmRidGJ0YoRidGJkYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzOTQyOTEyMjI1OTk4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQlInlHIiIhIiIiISIiRiZGJ0YnRiksJiokRiYiIiNGKCEiJUYoRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzOTQyOTEyMjI2OTU4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjMSIlIiUsJiUieEciIiJGKEYoIiIhJSJ5R0YpRilGJiEiI0YqLCZGJ0YoISIiRihGKUYqRitGKUYsRilGKkYl