What Does the Height of an Algebraic Equation Mean?

“Height of an algebraic equation” is not one universally fixed term. Most often, it measures the size of the coefficients in a chosen polynomial equation. In number theory, however, height often means an invariant of an algebraic number, such as one of the roots. These meanings are related but not interchangeable.

Height as coefficient size

Suppose an equation is given by a polynomial with integer coefficients,

$$P(x)=a_nx^n+\cdots+a_1x+a_0.$$

A common definition of its (naive) coefficient height is

$$H(P)=\max_{0\leq i\leq n}|a_i|.$$

Many authors instead use the logarithmic coefficient height, $h(P)=\log H(P)$. Check the author’s convention: some sources call the maximum itself “height,” while others reserve “height” for its logarithm.

For a polynomial with rational coefficients, first multiply by a common denominator to get integer coefficients, then divide all coefficients by their greatest common divisor. This gives a primitive integer polynomial, unique up to an overall sign. Its largest absolute coefficient is a standard way to define the coefficient height of the rational polynomial.

For example, the equation

$$6x^2-10x+4=0$$

can be divided by $2$ to obtain the primitive polynomial $3x^2-5x+2$. Its coefficient height is

$$H(P)=\max(3,5,2)=5,$$

and its logarithmic coefficient height is $\log 5$. Without the primitive normalization, multiplying the equation by any nonzero integer would change the apparent coefficient size, even though the solutions remain the same.

Height of a root

A different common meaning is the absolute logarithmic Weil height of an algebraic number $\alpha$. Let its primitive irreducible polynomial over the integers be

$$a\prod_{i=1}^{d}(x-\alpha_i),$$

where $a>0$ is the leading coefficient, $d$ is the degree, and the $\alpha_i$ are all its complex conjugates. Then

$$h(\alpha)=\frac{1}{d}\left(\log a+\sum_{i=1}^{d}\log\max(1,|\alpha_i|)\right).$$

This height is attached to the number, not just to whichever equation someone uses to describe it. Its multiplicative version is $H(\alpha)=e^{h(\alpha)}$.

For the rational number $\alpha=2/3$, the primitive minimal polynomial is $3x-2$. There is one conjugate, $2/3$, so

$$h(2/3)=\log 3+\log\max(1,2/3)=\log 3.$$

Thus $H(2/3)=3$. The root’s height happens to be easy to read from its degree-one minimal polynomial, but this does not mean the two definitions are the same in general.

For example, $\sqrt{2}$ has minimal polynomial $x^2-2$ and conjugates $\sqrt{2}$ and $-\sqrt{2}$. Therefore

$$h(\sqrt{2})=\frac{1}{2}\left(\log\sqrt{2}+\log\sqrt{2}\right)=\frac{1}{2}\log 2.$$

The polynomial’s coefficient height is $2$, whose logarithm is $\log 2$, not $\frac{1}{2}\log 2$. This shows why you need to identify which height is being discussed.

Why context matters

For an equation in several variables, a common coefficient-height convention for an integer polynomial is still the maximum absolute value of its coefficients. For a homogeneous polynomial, the same idea applies to its coefficients. But the height of a geometric object defined by the equation—such as a curve or a projective variety—may be defined differently, and it is not automatically the coefficient height.

The equation itself may also have multiple forms. Multiplying by a nonzero scalar does not change its solutions, so a normalization is needed to define coefficient height consistently. More broadly, changing the polynomial or using a different set of equations can describe the same solution set while giving different coefficient sizes. A height of a polynomial is therefore not automatically an invariant of the set it defines.

Common mistakes

• Assuming “height” always means degree. Degree counts the highest exponent; coefficient height measures coefficient size.
• Forgetting normalization. The coefficient maximum changes if you multiply a polynomial by a constant. For rational coefficients, use the primitive integer form under the stated convention.
• Confusing polynomial height with root height. The algebraic-number height uses the minimal polynomial and all its conjugates, with a degree normalization.
• Assuming every source uses the same convention. Check whether “height” means a maximum, its logarithm, or a more specialized arithmetic or geometric height.

When a text gives a formula or definition, use that definition over the generic phrase “height of the equation.” If it does not, the safest interpretation is coefficient size after normalization, while noting that the intended meaning may be the height of a root.