Height function

A height function measures arithmetic complexity rather than physical size. For a reduced rational number $a/b$, its multiplicative height is $H(a/b)=\max (|a|,|b|)$, and its logarithmic height is $h(a/b)=\log H(a/b)$. Thus $1001/1000$ is close to one in real size but has height $1001$.

For an algebraic number $\alpha$ of degree $d$, let its primitive irreducible integer polynomial have leading coefficient $a_d$ and complex roots ${\alpha }_1,\dots ,{\alpha }_d$. The Absolute logarithmic Weil height is $d^{-1}(\log |a_d|+{\sum }_i\log \max (1,|{\alpha }_i|))$. Bounding both height and degree leaves finitely many possibilities, making heights useful in Diophantine equations and finiteness arguments.

For a symmetric ample line bundle $L$ on an abelian variety, the canonical height is ${\hat{h}}_L(P)={\lim }_{n\to \infty }{4}^{-n}{h}_L([2^n]P)$. It is quadratic and differs from the associated logarithmic Weil height by a bounded function. The canonical height of an elliptic curve specializes this construction to a one-dimensional abelian variety.

On a projective variety over a number field, a very ample line bundle gives a logarithmic projective height by an embedding. Tensor products correspond to sums by the Segre embedding, and a sufficiently ample twist expresses any line bundle as a difference of very ample ones. This defines its height modulo bounded functions. It is additive and functorial under morphism pullback. An ample bundle has the Northcott theorem finiteness property for bounded height and bounded point degree; a trivial bundle need not. The product formula makes these absolute heights independent of scalar coordinates and of the number field used to compute them.

For an algebraic number $\alpha$ in a number field $L$, the absolute multiplicative Weil height isIt is independent of the field $L$ containing $\alpha$, satisfies $H({\alpha }^n)=H(\alpha {)}^{|n|}$ for nonzero $\alpha$ and every integer $n$, and obeys $H(\alpha +\beta )\le 2H(\alpha )H(\beta )$.

For coprime integers $p,q$ with $q>0$,This follows directly from the real and p-adic absolute values, or from the height-Mahler measure formula for the primitive polynomial $qX-p$.

The Absolute logarithmic Weil height is the logarithm $h(\alpha )=\log H(\alpha )$ of the absolute multiplicative height of an algebraic number. If its primitive integer minimal polynomial is $f(X)=a_d{\prod }_{i=1}^d(X-{\alpha }_i)$, thenThus it is the logarithm of the Mahler measure divided by the degree.

For a reduced rational $a/b$, $h(a/b)=\log \max (|a|,|b|)$. It satisfies $h({\alpha }^m)=|m|h(\alpha )$ for nonzero $\alpha$ and integral $m$. Unlike a single absolute value, it records denominator complexity and the behavior of all conjugates, making it suitable for arithmetic bounds that should not depend on a chosen embedding.

For a nonzero vector $\mathbf{a}=(a_0,\dots ,a_n)$ over a number field $K$, its absolute projective height isThe product formula makes this unchanged by multiplying all coordinates by the same nonzero scalar. The projective height of a linear form is the projective height of its coefficient vector.

The naive height of a polynomial is the largest absolute value of its coefficients. For a primitive polynomial in $\mathbb{Z}[X_1,\dots ,X_n]$, this equals the projective height of its coefficient vector.

If the primitive minimal polynomial of an algebraic number $\alpha$ has degree $d$, then

Let $P,Q\in \mathbb{Z}[X_1,\dots ,X_k]$ have degree at most $n_j$ in $X_j$. Thenwhenever the quotient is defined. At each non-Archimedean place, the integral coefficients contribute at most one; at the Archimedean places, the triangle inequality contributes the polynomial length. Multiplying these local estimates and using the product formula gives the claim.

If $\gamma \ne 0$ is algebraic of degree at most $D$, then every embedding satisfiesIndeed, $H({\gamma }^{-1})=H(\gamma )$ and the contribution of one Archimedean place to the height bounds $|\gamma {|}^{-1/D}$.

There are only finitely many algebraic numbers of bounded degree and bounded Absolute multiplicative Weil height. The finite set can be enumerated effectively from the bounded coefficients of their primitive minimal polynomials.

The degree and Absolute multiplicative Weil height bounds are both essential.

For the primitive minimal polynomial $a_d{\prod }_j(T-{\alpha }_j)$, the height-Mahler measure formula gives $H(\alpha {)}^d=a_d{\prod }_j\max (1,|{\alpha }_j|)$. Expanding the product bounds every integer coefficient by $2^D{B}^D$. There are finitely many bounded coefficient lists and each has finitely many roots, proving finiteness of the algebraic numbers. On a projective coordinate chart, every coordinate ratio has height at most the point's projective height; applying the scalar bound proves the projective Northcott theorem too. A degree bound is essential, as the infinitely many roots of unity all have height one.