Résultantes

Résultantes, in mathematics, specifically in the context of polynomial algebra, refers to a concept used to determine conditions for which two polynomials share a common root. It is a symmetric polynomial in the roots of the given polynomials.

The resultant of two polynomials, say $A(x)$ and $B(x)$, can be defined as a value that is zero if and only if $A(x)$ and $B(x)$ have a common root in an algebraically closed field. This value can be computed using the Sylvester matrix, whose determinant is the resultant. If $A(x)$ has degree $m$ and $B(x)$ has degree $n$, the Sylvester matrix is of size $(m+n) \times (m+n)$.

The resultant has several important properties. For instance, it is zero if and only if the polynomials have a common root. It is multiplicative, meaning that the resultant of the product of two polynomials with a third is the product of their individual resultants. This property makes it useful in various algebraic manipulations and in the study of systems of polynomial equations.

Beyond its theoretical significance, the resultant has practical applications in areas such as algebraic geometry, computer algebra systems for solving polynomial equations, and in the analysis of dynamical systems where conditions for bifurcations can be related to resultants. It provides a powerful tool for understanding the relationships between polynomial roots and their commonalities without explicitly finding the roots themselves.