Theory of equations/Resultants and discriminants

Resultants and discriminants are algebraic tools for studying how two polynomials share roots, and how the roots of a single polynomial collide (become equal). They are widely used in elimination theory, algebraic geometry, and the theory of equations. __TOC__ Intuitive picture Imagine two polynomials in one variable, as two “curves” on the -plane (if we rewrite them as and ). Where they intersect, they share an -value that is a root of both polynomials. The resultant is a number (built from the coefficients of and ) which is zero exactly when such a common root exists. For a single polynomial, the discriminant is a number that vanishes precisely when the polynomial has a repeated root. It is therefore a convenient way to detect whether the graph of the polynomial just touches the -axis or crosses it. Resultant of two polynomials Let be polynomials over a commutative ring (often a field) with . Denote their roots in some extension field by Definition via roots The resultant of and , written , is From this expression one sees immediately: if and only if for some , i.e. and have a common root. The resultant is a polynomial expression in the coefficients , and therefore can be computed without ever explicitly finding the roots. Definition via Sylvester matrix In practice, the resultant is computed as a determinant. The Sylvester matrix of and is an matrix built by stacking shifted rows of coefficients: the first rows contain the coefficients of shifted to the right, the next rows contain the coefficients of shifted to the right. For example, if then the Sylvester matrix is The resultant is defined by This determinant has the same value (up to a nonzero factor depending only on ) as the product form given above, and in particular it vanishes if and only if and have a common root. Example: eliminating a variable Suppose we want to eliminate from the system Here so they share the root . The resultant must therefore be zero. Using the Sylvester matrix, so A direct computation shows that , confirming the existence of a common root without solving either equation explicitly. Resultants are often used in more complicated problems to eliminate a variable and derive a single equation relating parameters appearing in the coefficients. Discriminant of a polynomial Let The discriminant of , denoted , is defined as Important properties: if and only if at least two roots coincide, i.e. has a multiple root. Over the real numbers, the sign of contains information about how many real roots has (for low degrees, especially quadratics and cubics). There is a close link between discriminant and resultant: Here is the derivative of . Since common roots of and correspond exactly to multiple roots of , the vanishing of this resultant detects repeated roots. Quadratic case For one computes The familiar expression therefore measures how far apart the two roots are. When , they collide into a repeated root; when (over the reals) the roots are distinct real numbers; and when it is negative the roots are complex conjugates. Cubic case (outline) For a cubic the discriminant is Although this expression is more complicated, it plays the same role: it vanishes exactly when the cubic has a multiple root, and its sign over the reals gives information about the number of real roots. Geometric meaning The resultant of and can be viewed as the signed “product of vertical distances” between their graphs taken over all pairs of roots. When two roots coincide, a factor vanishes and the resultant is zero. The discriminant of measures how “separated” its roots are. Very small discriminant (in absolute value) means the polynomial has roots that are very close to each other, while a large discriminant means they are well separated. These ideas are important in numerical analysis, where they help to understand the stability of roots under small changes of the coefficients.