General Theory of Polynomials in Several Variables

General Theory of Polynomials in Several Variables#

Comments on nomenclature

In the general theory of polynomials (algebraic geometry), a form of degree \(n\) refers to a polynomial homogeneous in that degree. This means that the polynomial only contains degree-\(n\) terms. In this context, the term form has a technical meaning. On the other hand, a polynomial equation in degree \(n\) may contain lower order terms. We may speak of “the general form of” an equation, but this sense of the term form is not special.

general \(n\)-th degree polynomial equation in \(k\) variables

  • first-degree = linear

  • second-degree = quadratic

  • third-degree = cubic

  • fourth-degree = quartic

  • fifth-degree = quintic

Degree-one polynomial equations in several variables#

Linear in a single variable

The general form of a linear equation in a single variables \(x\) is

\(Ax + B = 0\) where \(A \ne 0\)

\( \begin{aligned} Ax = -B \iff x = -\frac{B}{A} \end {aligned} \)

Linear in two variables

The general form of a linear equation in two variables \(x, y\) is

\(Ax + By + C = 0\) where \(A \ne 0 \lor B \ne 0\)

From here we derive the standard form of a line (in which this \(C\) is the negative of the \(C\) above)

\(Ax + By = C\)

From here we derive the slope-intercept form of a line \(y = mx + b\).

\( \begin{aligned} Ax + By = C \iff By = -Ax + C \iff y = \underset{m}{\boxed{-\frac{A}{B}}}x + \underset{b}{\boxed{\frac{C}{B}}} \end {aligned} \)

From here we derive the point-slope form of a line \(y - y_0 = m(x - x_0)\) for a given point \((x_0, y_0)\) and slope \(m\).

\(y = mx + b \iff b = -mx + y\) for any \((x, y)\). Let’s say \(x = x_0\) and \(y = y_0\). Then \(b = -mx_0 + y_0\). Now we can substitute this back into the original equation.

\(y = mx + b = mx + (-mx_0 + y_0) = m(x - x_0) + y_0\)

Linear in several variables

The general form of a linear equation in several variables \(x_1, \dotsc, x_n\) is

\(a_1 x_1 + a_2 x_2 + \dotsb + a_n x_n + b = 0\) where \(a_1 \ne 0 \lor a_2 \ne 0 \lor \dotsb \lor a_n \ne 0\)

From here we derive linear forms in higher dimensions (the plane in three dimensions and the hyperplane in greater-than-three dimensions).

Degree-two polynomial equations in several variables#

General quadratic in one variable

The general form of a quadratic equation in a single variable \(x\) is

\(Ax^2 + Bx + C = 0\) where \(A \ne 0\)

General quadratic in two variables

The general form of a quadratic equation in two variables \(x, y\)

with cross terms is

\(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\) where \(A\ne0\lor B\ne0\lor C\ne0\)

without cross terms is

\(Ax^2+By^2+Cx+Dy+E=0\) where \(A\ne0\lor B\ne0\)

In classical analytic geometry, this is the general equation of a conic section.

Discriminant#

For a degree-\(n\) polynomial in a single variable

\(p(x) = a_n x^n + \dotsb + a_1 x + a_0\)

with roots \(r_1, \dotsc, r_n\) the discriminant is defined as

\( \begin{aligned} \text{Disc}(p) = a_n^{2n - 2} \prod_{i \lt j} (r_i - r_j)^2 \end {aligned} \)

For the degree-two polynomial equation in a single variable

\(p(x) = a x^2 + b x + c\)

this gives

\(\text{Disc}(p) = b^2 - 4ac\)

This tells us about the nature of the roots of the polynomial

\( \begin{aligned} \text{Disc}(p) &\gt 0 && \text{two distinct real roots} \\ \text{Disc}(p) &= 0 && \text{one repeated real root} \\ \text{Disc}(p) &\lt 0 && \text{two complex roots} \\ \end {aligned} \)

In the context of degree-\(n\) polynomial equations in a single variable, the discriminant is defined as a function of the coefficients. The vanishing of the discriminant is an indicator of degeneracy, and the condition of degeneracy is the multiplicity of the roots.

Terms#

  • [ w ] Abel-Ruffini Theorem

  • [ w ] Algebraic Equation (Polynomial Equation)

  • [ w ] Algebraic Expression

  • [ w ] Algebraic Solution

  • [ w ] Bring Radical

  • [ w ] Determinant

  • [ w ] Discriminant

  • [ w ] Equation, Algebraic (Polynomial)

  • [ w ] Equation, Polynomial (Algebraic)

  • [ w ] Form (or Homogeneous Polynomial or Quantic)

  • [ w ] Galois Theory

  • [ w ] Homogeneous Coordinates

  • [ w ] Homogeneous Polynomial (or Form or Quantic)

  • [ w ] Hypersurface

  • [ w ] Irreducible Polynomial

  • [ w ] Linear Form

  • [ w ] Polynomial, Irreducible

  • [ w ] Polynomial Equation (Algebraic Equation)

  • [ w ] Quadratic Form

  • [ w ] Quadratic Formula

  • [ w ] Quadric

  • [ w ] Quantic (or Form or Homogeneous Polynomial)

  • [ w ] Radical Symbol

  • [ w ] Resultant

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