5.2 Resultants and discriminants

Resultants and discriminants are powerful tools in algebraic geometry. They help us understand how polynomials intersect and behave, without solving equations directly. These concepts are key to elimination theory, allowing us to simplify systems of equations and study their properties.

By computing resultants and discriminants, we can find common roots, analyze singularities, and eliminate variables. This connects to the broader chapter by showing how we can extract valuable information from polynomial systems without explicit solutions.

Resultants and Discriminants for Polynomials

Definition and Properties of Resultants

• The resultant of two univariate polynomials $f(x)$ and $g(x)$ is a polynomial expression in their coefficients that vanishes if and only if $f$ and $g$ have a common root
• For multivariate polynomials $f(x_1, ..., x_n)$ and $g(x_1, ..., x_n)$, the resultant with respect to a variable $x_i$ is a polynomial in the remaining variables that vanishes if and only if $f$ and $g$ have a common root when considered as polynomials in $x_i$
  • Treats the coefficients of $f$ and $g$ as polynomials in the other variables
  • Allows for elimination of a variable from a system of polynomial equations
• Resultants are symmetric, meaning $Res(f, g) = Res(g, f)$
• The resultant of $f$ and $g$ is divisible by their greatest common divisor (GCD)

Definition and Properties of Discriminants

• The discriminant of a univariate polynomial $f(x)$ is a polynomial expression in its coefficients that vanishes if and only if $f$ has a multiple root
  • Provides information about the nature of the roots without explicitly solving the equation
  • If the discriminant is positive, $f$ has distinct real roots; if it is zero, $f$ has a multiple real root; and if it is negative, $f$ has complex conjugate roots
• For a multivariate polynomial $f(x_1, ..., x_n)$, the discriminant with respect to a variable $x_i$ is a polynomial in the remaining variables that vanishes if and only if $f$ has a multiple root when considered as a polynomial in $x_i$
  • Treats the coefficients of $f$ as polynomials in the other variables
  • Gives information about the singularities of the hypersurface defined by $f$ when projected onto the hyperplane $x_i = 0$

Computing Resultants and Discriminants

Sylvester's Matrix and Determinant

• Sylvester's matrix for two univariate polynomials $f(x)$ and $g(x)$ of degrees $m$ and $n$, respectively, is an $(m+n) \times (m+n)$ matrix constructed from their coefficients
  • The resultant of $f$ and $g$ is the determinant of Sylvester's matrix
  • Example: For $f(x) = a_2x^2 + a_1x + a_0$ and $g(x) = b_1x + b_0$, Sylvester's matrix is $\begin{bmatrix} a_2 & a_1 & a_0 & 0 \\ 0 & a_2 & a_1 & a_0 \\ b_1 & b_0 & 0 & 0 \\ 0 & b_1 & b_0 & 0 \end{bmatrix}$
• The discriminant of a univariate polynomial $f(x)$ can be computed using Sylvester's matrix of $f$ and its derivative $f'$
  • The discriminant is the resultant of $f$ and $f'$ divided by the leading coefficient of $f$

Bezout's Determinant and Subresultants

• Bezout's determinant is another method for computing the resultant of two univariate polynomials $f(x)$ and $g(x)$
  • Involves constructing a matrix using the coefficients of $f$, $g$, and their derivatives, and then calculating the determinant of this matrix
  • Can be more efficient than Sylvester's matrix for high-degree polynomials
• Subresultant sequences provide an efficient method for computing resultants
  • Based on the Euclidean algorithm for polynomial division
  • Can also be used to compute the GCD of two polynomials
  • Example: The subresultant sequence of $f(x) = a_2x^2 + a_1x + a_0$ and $g(x) = b_1x + b_0$ is $\begin{aligned} S_0 &= a_2x^2 + a_1x + a_0 \\ S_1 &= b_1x + b_0 \\ S_2 &= (a_1b_0 - a_0b_1)x + (a_2b_0 - a_0b_1) \end{aligned}$

Multivariate Resultants and Discriminants

• For multivariate polynomials, resultants and discriminants can be computed by treating them as univariate polynomials in one variable while considering the coefficients as polynomials in the remaining variables
  • The computation then proceeds using Sylvester's matrix, Bezout's determinant, or subresultant sequences
  • Example: For $f(x, y) = a_2(y)x^2 + a_1(y)x + a_0(y)$ and $g(x, y) = b_1(y)x + b_0(y)$, the resultant with respect to $x$ is a polynomial in $y$ obtained by computing the resultant of $f$ and $g$ as univariate polynomials in $x$ with coefficients in $\mathbb{C}[y]$
• Gröbner bases can be used to compute multivariate resultants and discriminants
  • Provide a systematic way to eliminate variables from a system of polynomial equations
  • Can be more efficient than direct computation using Sylvester's matrix or Bezout's determinant

Geometric Interpretation of Resultants and Discriminants

Intersection of Plane Curves

• For two plane curves defined by polynomial equations $f(x, y) = 0$ and $g(x, y) = 0$, the resultant of $f$ and $g$ with respect to either $x$ or $y$ vanishes if and only if the curves have a common point of intersection
  • The resultant with respect to $x$ gives a polynomial in $y$ whose roots correspond to the $y$-coordinates of the intersection points
  • The resultant with respect to $y$ gives a polynomial in $x$ whose roots correspond to the $x$-coordinates of the intersection points
  • Example: The curves $y = x^2$ and $y = x + 1$ intersect at $(0, 1)$ and $(-1, 0)$, which can be found by computing the resultant of the polynomials $f(x, y) = y - x^2$ and $g(x, y) = y - x - 1$ with respect to either $x$ or $y$

Singularities of Plane Curves

• The discriminant of a plane curve $f(x, y) = 0$ vanishes if and only if the curve has a singular point where the partial derivatives of $f$ vanish simultaneously
  • A singular point can be a node (two branches intersecting), a cusp (two branches meeting tangentially), or an isolated point
  • The discriminant with respect to $x$ gives a polynomial in $y$ whose roots correspond to the $y$-coordinates of the singular points
  • The discriminant with respect to $y$ gives a polynomial in $x$ whose roots correspond to the $x$-coordinates of the singular points
  • Example: The curve $y^2 = x^3$ has a cusp at $(0, 0)$, which can be found by computing the discriminant of the polynomial $f(x, y) = y^2 - x^3$ with respect to either $x$ or $y$

Intersection and Singularities in Higher Dimensions

• In higher dimensions, the resultant of two hypersurfaces $f(x_1, ..., x_n) = 0$ and $g(x_1, ..., x_n) = 0$ with respect to a variable $x_i$ vanishes if and only if the hypersurfaces intersect when projected onto the hyperplane defined by $x_i = 0$
  • The resultant gives a polynomial in the remaining variables whose vanishing set corresponds to the projection of the intersection
• The discriminant of a hypersurface $f(x_1, ..., x_n) = 0$ with respect to a variable $x_i$ vanishes if and only if the hypersurface has a singularity when projected onto the hyperplane defined by $x_i = 0$
  • The discriminant gives a polynomial in the remaining variables whose vanishing set corresponds to the projection of the singular locus
  • Example: The surface $z = x^2 + y^2$ has a singularity at $(0, 0, 0)$, which can be found by computing the discriminant of the polynomial $f(x, y, z) = z - x^2 - y^2$ with respect to either $x$, $y$, or $z$

Applications of Resultants and Discriminants

Solving Polynomial Equations

• Resultants can be used to eliminate a variable from a system of polynomial equations, reducing the problem to a single equation in fewer variables
  • This technique is known as resultant-based variable elimination
  • Example: To solve the system $\begin{cases} x^2 + y^2 = 1 \\ x + y = 1 \end{cases}$, compute the resultant of the polynomials $f(x, y) = x^2 + y^2 - 1$ and $g(x, y) = x + y - 1$ with respect to $x$ to obtain a quadratic equation in $y$, which can be solved to find the $y$-coordinates of the solutions
• Discriminants can be used to determine the nature of the roots of a polynomial equation without explicitly solving it
  • Example: The discriminant of the quadratic equation $ax^2 + bx + c = 0$ is $\Delta = b^2 - 4ac$, which determines the nature of the roots: if $\Delta > 0$, there are two distinct real roots; if $\Delta = 0$, there is a double real root; and if $\Delta < 0$, there are two complex conjugate roots

Analyzing Algebraic Varieties

• Resultants and discriminants can be used to study the geometry of algebraic varieties, which are sets of points defined by polynomial equations
  • The resultant of two varieties gives information about their intersection
  • The discriminant of a variety gives information about its singularities
• In combination with other algebraic techniques, such as Gröbner bases and real root isolation, resultants and discriminants can be used to solve systems of polynomial equations and analyze their solutions
  • Gröbner bases provide a systematic way to eliminate variables and simplify polynomial systems
  • Real root isolation algorithms, such as Sturm sequences or Descartes' rule of signs, can be used to locate the real solutions of a polynomial equation
  • Example: To find the intersection points of the curves $y = x^2$ and $y = x + 1$, compute the resultant of the polynomials $f(x, y) = y - x^2$ and $g(x, y) = y - x - 1$ with respect to $y$ to obtain a polynomial in $x$, and then use real root isolation to find its real roots, which correspond to the $x$-coordinates of the intersection points

Optimization and Constraint Satisfaction

• Resultants and discriminants can be applied to solve optimization problems involving polynomial constraints
  • The resultant of the objective function and the constraint polynomials can be used to eliminate variables and reduce the problem to a single equation
  • The discriminant of the reduced equation can provide information about the existence and nature of the optimal solutions
  • Example: To minimize $x^2 + y^2$ subject to the constraint $xy = 1$, compute the resultant of the polynomials $f(x, y) = x^2 + y^2$ and $g(x, y) = xy - 1$ with respect to $y$ to obtain a polynomial in $x$, and then find its minimum value using calculus or numerical methods
• In constraint satisfaction problems, resultants and discriminants can be used to determine the feasibility of polynomial constraints and to find solutions that satisfy them
  • The resultant of the constraint polynomials can be used to check if they have a common solution
  • The discriminant of the constraint polynomials can be used to identify singular solutions or solutions with specific properties
  • Example: To find points $(x, y)$ that satisfy the constraints $\begin{cases} x^2 + y^2 \leq 1 \\ xy \geq 0 \end{cases}$, compute the resultant of the polynomials $f(x, y) = x^2 + y^2 - 1$ and $g(x, y) = xy$ with respect to $y$ to obtain a polynomial in $x$, and then find its real roots in the interval $[-1, 1]$ using real root isolation