Multivariable Boyd–Lawton Theorem

• The multivariable Boyd–Lawton theorem is a set of limit formulas that recover Mahler-type invariants of multivariate polynomials through increasingly generic univariate or torus homomorphism substitutions.
• It employs classical techniques like Jensen’s formula and torus integrals alongside higher-dimensional generalizations, using lattice invariants and Fourier analysis to control error estimates.
• The theorem extends to generalized, multiple, higher, and dynamical Mahler measures, linking analytic, number theoretic, and operator-algebraic frameworks to address approximation and Lehmer-type problems.

The multivariable Boyd–Lawton theorem is a family of limit formulas asserting that Mahler-type invariants of a polynomial in several variables can be recovered from Mahler-type invariants of lower-dimensional monomial specializations. In its classical form, it expresses the Mahler measure of a nonzero multivariate polynomial as the limit of Mahler measures of univariate polynomials obtained by substitutions of the form $x_j=t^{r_j}$ under a genericity condition excluding short integer relations among the exponents. Subsequent work extends this principle in two directions: from univariate specializations to arbitrary homomorphisms $T^m\to T^n$, and from ordinary Mahler measure to generalized, multiple, higher, dynamical, and other related measures (Issa et al., 2012, Brunault et al., 2022, Aitken et al., 8 Aug 2025).

1. Classical formulation

For a nonzero $n$-variable polynomial $P(x_1,\dots,x_n)\in\mathbb{C}[x_1,\dots,x_n]$, the Mahler measure is

$$m(P)=\frac{1}{(2\pi)^n}\int_{\mathbb{T}^n}\log|P(x_1,\dots,x_n)|\,dx_1\cdots dx_n,$$

where $$\mathbb{T}^n=\{(z_1,\dots,z_n)\in\mathbb{C}^n:|z_1|=\cdots=|z_n|=1\}$$. In one variable, if $P(x)=a\prod_i(x-\alpha_i)$, Jensen’s formula gives

$m(P)=\log|a|+\sum_i\max\{0,\log|\alpha_i|\}.$

The classical Boyd–Lawton theorem considers, for $\mathbf r=(r_1,\dots,r_n)\in\mathbb Z_{>0}^n$, the specialization

$$P_{\mathbf r}(x)=P(x^{r_1},\dots,x^{r_n})\in\mathbb C[x].$$

Its genericity parameter is

$T^m\to T^n$0

The theorem states that

$T^m\to T^n$1

Thus a genuinely $T^m\to T^n$2-variable torus integral is obtained as a limit of one-variable Mahler measures along increasingly generic monomial curves $T^m\to T^n$3 (Issa et al., 2012).

2. Genericity, torus maps, and homomorphism language

The theorem is naturally interpreted through homomorphisms of compact tori. In the classical one-parameter setting, $T^m\to T^n$4 defines

$T^m\to T^n$5

and the condition $T^m\to T^n$6 means that $T^m\to T^n$7 admits no short nontrivial integral relation. This is the Diophantine mechanism behind the approximation.

A higher-dimensional formulation replaces exponent vectors by integer matrices. For $T^m\to T^n$8, one defines

$T^m\to T^n$9

where $n$0 is the monomial map determined by $n$1. The relevant lattice invariant is

$n$2

with $n$3 when $n$4. The push-forward of Haar measure on $n$5 under $n$6 is a probability measure $n$7 on $n$8, and

$n$9

The Fourier coefficients of $P(x_1,\dots,x_n)\in\mathbb{C}[x_1,\dots,x_n]$0 are supported on $P(x_1,\dots,x_n)\in\mathbb{C}[x_1,\dots,x_n]$1; when $P(x_1,\dots,x_n)\in\mathbb{C}[x_1,\dots,x_n]$2, $P(x_1,\dots,x_n)\in\mathbb{C}[x_1,\dots,x_n]$3 converges weakly to Haar measure on $P(x_1,\dots,x_n)\in\mathbb{C}[x_1,\dots,x_n]$4 (Brunault et al., 2022).

A parallel but transposed notation appears in the homomorphism-based framework of Aitken–Ayers–Smith. There every continuous homomorphism $P(x_1,\dots,x_n)\in\mathbb{C}[x_1,\dots,x_n]$5 is $P(x_1,\dots,x_n)\in\mathbb{C}[x_1,\dots,x_n]$6 for a unique integer matrix $P(x_1,\dots,x_n)\in\mathbb{C}[x_1,\dots,x_n]$7, and the associated Boyd height is

$P(x_1,\dots,x_n)\in\mathbb{C}[x_1,\dots,x_n]$8

In that language, $P(x_1,\dots,x_n)\in\mathbb{C}[x_1,\dots,x_n]$9 exactly when $$m(P)=\frac{1}{(2\pi)^n}\int_{\mathbb{T}^n}\log|P(x_1,\dots,x_n)|\,dx_1\cdots dx_n,$$0 has rank $$m(P)=\frac{1}{(2\pi)^n}\int_{\mathbb{T}^n}\log|P(x_1,\dots,x_n)|\,dx_1\cdots dx_n,$$1, equivalently when $$m(P)=\frac{1}{(2\pi)^n}\int_{\mathbb{T}^n}\log|P(x_1,\dots,x_n)|\,dx_1\cdots dx_n,$$2 is surjective (Aitken et al., 8 Aug 2025).

3. Arbitrary multivariable monomial substitutions

The higher-dimensional extension proved by Brunault, Guilloux, Mehrabdollahei, and Pengo allows arbitrary sequences of integer matrices $$m(P)=\frac{1}{(2\pi)^n}\int_{\mathbb{T}^n}\log|P(x_1,\dots,x_n)|\,dx_1\cdots dx_n,$$3, with the number of rows $$m(P)=\frac{1}{(2\pi)^n}\int_{\mathbb{T}^n}\log|P(x_1,\dots,x_n)|\,dx_1\cdots dx_n,$$4 varying. If $$m(P)=\frac{1}{(2\pi)^n}\int_{\mathbb{T}^n}\log|P(x_1,\dots,x_n)|\,dx_1\cdots dx_n,$$5 is a Laurent polynomial and $$m(P)=\frac{1}{(2\pi)^n}\int_{\mathbb{T}^n}\log|P(x_1,\dots,x_n)|\,dx_1\cdots dx_n,$$6, then

$$m(P)=\frac{1}{(2\pi)^n}\int_{\mathbb{T}^n}\log|P(x_1,\dots,x_n)|\,dx_1\cdots dx_n,$$7

This is a genuine multivariable Boyd–Lawton theorem: the approximants need not be univariate, and the monomial substitutions may come from arbitrary torus homomorphisms rather than a single monomial curve (Brunault et al., 2022).

The same work gives an explicit error term. If $$m(P)=\frac{1}{(2\pi)^n}\int_{\mathbb{T}^n}\log|P(x_1,\dots,x_n)|\,dx_1\cdots dx_n,$$8 has exactly $$m(P)=\frac{1}{(2\pi)^n}\int_{\mathbb{T}^n}\log|P(x_1,\dots,x_n)|\,dx_1\cdots dx_n,$$9 nonzero coefficients and Newton polytope diameter $$\mathbb{T}^n=\{(z_1,\dots,z_n)\in\mathbb{C}^n:|z_1|=\cdots=|z_n|=1\}$$0, and if $$\mathbb{T}^n=\{(z_1,\dots,z_n)\in\mathbb{C}^n:|z_1|=\cdots=|z_n|=1\}$$1 with

$$\mathbb{T}^n=\{(z_1,\dots,z_n)\in\mathbb{C}^n:|z_1|=\cdots=|z_n|=1\}$$2

then

$$\mathbb{T}^n=\{(z_1,\dots,z_n)\in\mathbb{C}^n:|z_1|=\cdots=|z_n|=1\}$$3

When $$\mathbb{T}^n=\{(z_1,\dots,z_n)\in\mathbb{C}^n:|z_1|=\cdots=|z_n|=1\}$$4 has no zeros on $$\mathbb{T}^n=\{(z_1,\dots,z_n)\in\mathbb{C}^n:|z_1|=\cdots=|z_n|=1\}$$5, the convergence becomes exponentially fast: there exist $$\mathbb{T}^n=\{(z_1,\dots,z_n)\in\mathbb{C}^n:|z_1|=\cdots=|z_n|=1\}$$6 and $$\mathbb{T}^n=\{(z_1,\dots,z_n)\in\mathbb{C}^n:|z_1|=\cdots=|z_n|=1\}$$7 such that

$$\mathbb{T}^n=\{(z_1,\dots,z_n)\in\mathbb{C}^n:|z_1|=\cdots=|z_n|=1\}$$8

under the stated lower bound on $$\mathbb{T}^n=\{(z_1,\dots,z_n)\in\mathbb{C}^n:|z_1|=\cdots=|z_n|=1\}$$9 in terms of $P(x)=a\prod_i(x-\alpha_i)$0 (Brunault et al., 2022).

This formulation clarifies that the decisive parameter is not the number of substitution variables but the absence of short integral relations in the exponent lattice. It also places the theorem in a measure-theoretic setting: $P(x)=a\prod_i(x-\alpha_i)$1 is singular on the toric zero locus, yet weak convergence of $P(x)=a\prod_i(x-\alpha_i)$2 together with uniform $P(x)=a\prod_i(x-\alpha_i)$3-control still suffices to pass to the limit (Brunault et al., 2022).

4. Generalized, multiple, and higher Mahler measures

Issa and Lalín extended the classical Boyd–Lawton theorem from ordinary Mahler measure to three further torus integrals. For nonzero polynomials $P(x)=a\prod_i(x-\alpha_i)$4, they define the generalized Mahler measure

$P(x)=a\prod_i(x-\alpha_i)$5

the multiple Mahler measure

$P(x)=a\prod_i(x-\alpha_i)$6

and, when $P(x)=a\prod_i(x-\alpha_i)$7, the higher Mahler measure

$P(x)=a\prod_i(x-\alpha_i)$8

For the same specialization $P(x)=a\prod_i(x-\alpha_i)$9, they proved

$m(P)=\log|a|+\sum_i\max\{0,\log|\alpha_i|\}.$0

$m(P)=\log|a|+\sum_i\max\{0,\log|\alpha_i|\}.$1

and hence

$m(P)=\log|a|+\sum_i\max\{0,\log|\alpha_i|\}.$2

Their analytic input is control of the singularities of $m(P)=\log|a|+\sum_i\max\{0,\log|\alpha_i|\}.$3 near toric zeros via Lawton’s sublevel-set estimate and bounds for integrals over sets where $m(P)=\log|a|+\sum_i\max\{0,\log|\alpha_i|\}.$4 is small (Issa et al., 2012).

Aitken–Ayers–Smith then formulated an abstract framework of “Boyd–Lawton collections.” A collection $m(P)=\log|a|+\sum_i\max\{0,\log|\alpha_i|\}.$5 is required to consist of integrable functions, satisfy the one-variable Boyd–Lawton theorem, and be stable under composition with sufficiently high-height torus homomorphisms. They showed that continuous functions, ordinary Mahler integrands $m(P)=\log|a|+\sum_i\max\{0,\log|\alpha_i|\}.$6, generalized Mahler integrands $m(P)=\log|a|+\sum_i\max\{0,\log|\alpha_i|\}.$7, and product integrands $m(P)=\log|a|+\sum_i\max\{0,\log|\alpha_i|\}.$8 are all Boyd–Lawton collections. Consequently, if $m(P)=\log|a|+\sum_i\max\{0,\log|\alpha_i|\}.$9, then

$\mathbf r=(r_1,\dots,r_n)\in\mathbb Z_{>0}^n$0

with $\mathbf r=(r_1,\dots,r_n)\in\mathbb Z_{>0}^n$1 allowed to vary. For surjective $\mathbf r=(r_1,\dots,r_n)\in\mathbb Z_{>0}^n$2, equivalently $\mathbf r=(r_1,\dots,r_n)\in\mathbb Z_{>0}^n$3, the equality is exact rather than asymptotic. In particular, generalized and multiple higher Mahler measures admit full multivariable Boyd–Lawton theorems parallel to the classical one (Aitken et al., 8 Aug 2025).

5. Dynamical and nonclassical analogues

A dynamical analogue replaces the unit torus and Haar measure by the Julia set $\mathbf r=(r_1,\dots,r_n)\in\mathbb Z_{>0}^n$4 of a monic polynomial $\mathbf r=(r_1,\dots,r_n)\in\mathbb Z_{>0}^n$5 of degree $\mathbf r=(r_1,\dots,r_n)\in\mathbb Z_{>0}^n$6 and its equilibrium measure $\mathbf r=(r_1,\dots,r_n)\in\mathbb Z_{>0}^n$7. For $\mathbf r=(r_1,\dots,r_n)\in\mathbb Z_{>0}^n$8, the multivariable dynamical Mahler measure is

$\mathbf r=(r_1,\dots,r_n)\in\mathbb Z_{>0}^n$9

In two variables, the available Boyd–Lawton-type statement is a weak inequality: $$P_{\mathbf r}(x)=P(x^{r_1},\dots,x^{r_n})\in\mathbb C[x].$$0 The specialization here is the dynamical graph $$P_{\mathbf r}(x)=P(x^{r_1},\dots,x^{r_n})\in\mathbb C[x].$$1, not a monomial torus homomorphism. Equality is conjectured but not proved in that setting. The weak result is nevertheless sufficient, together with a dynamical Lehmer conjecture and a dynamical Kronecker lemma, for the classification of irreducible $$P_{\mathbf r}(x)=P(x^{r_1},\dots,x^{r_n})\in\mathbb C[x].$$2 with $$P_{\mathbf r}(x)=P(x^{r_1},\dots,x^{r_n})\in\mathbb C[x].$$3 under the exclusion of the power-map and Chebyshev cases (Carter et al., 2021).

A different analogue arises from the “alternative to Mahler measure.” For a univariate polynomial $$P_{\mathbf r}(x)=P(x^{r_1},\dots,x^{r_n})\in\mathbb C[x].$$4 of degree $$P_{\mathbf r}(x)=P(x^{r_1},\dots,x^{r_n})\in\mathbb C[x].$$5, $$P_{\mathbf r}(x)=P(x^{r_1},\dots,x^{r_n})\in\mathbb C[x].$$6 is the ratio $$P_{\mathbf r}(x)=P(x^{r_1},\dots,x^{r_n})\in\mathbb C[x].$$7, where $$P_{\mathbf r}(x)=P(x^{r_1},\dots,x^{r_n})\in\mathbb C[x].$$8 is the number of roots in the open unit disk, and Cauchy’s argument principle yields an integral formula. For a multivariate polynomial $$P_{\mathbf r}(x)=P(x^{r_1},\dots,x^{r_n})\in\mathbb C[x].$$9 with no zeros on $T^m\to T^n$00, one defines directional invariants

$T^m\to T^n$01

where $T^m\to T^n$02 is the degree in $T^m\to T^n$03. The resulting Boyd–Lawton-type theorem is

$T^m\to T^n$04

In this theory the limit is a sum of directional contributions rather than a single symmetric invariant; explicit evaluations include $T^m\to T^n$05 and $T^m\to T^n$06 (Stankov, 5 Feb 2025).

6. Arithmetic significance and broader context

The theorem occupies a central position at the interface of analysis, number theory, and, in some variants, dynamics. Classical Mahler measures of multivariate polynomials are known to produce special values of the Riemann zeta function and $T^m\to T^n$07-functions, and the Boyd–Lawton principle allows these multivariable quantities to be approximated by Mahler measures of lower-dimensional specializations. Issa–Lalín explicitly note analogous links for generalized, multiple, and higher Mahler measures, including connections to multiple sine functions, multiple zeta values, and Witten zeta values associated to Lie algebras (Issa et al., 2012).

The theorem also interacts with Lehmer-type problems. In the classical case, Boyd–Lawton implies that very small positive Mahler measure in several variables would force the existence of infinitely many univariate polynomials with equally small Mahler measure. The same propagation principle extends to generalized and multiple measures in Issa–Lalín’s setting (Issa et al., 2012). In the dynamical setting, the weak two-variable inequality plays an analogous role by transferring vanishing of a multivariable dynamical measure to vanishing of many one-variable dynamical measures, which can then be analyzed through canonical heights and preperiodic points (Carter et al., 2021).

The multivariable matrix formulation broadens the scope further. Brunault–Guilloux–Mehrabdollahei–Pengo connect these limit formulas to limit sets of Mahler measures, to previously studied convergent families such as the polynomials $T^m\to T^n$08, and to operator-algebraic contexts involving entropy and Fuglede–Kadison determinants. Their example

$T^m\to T^n$09

is obtained by representing $T^m\to T^n$10 as a specialization of a four-variable polynomial $T^m\to T^n$11 through matrices $T^m\to T^n$12 with $T^m\to T^n$13, and the same family admits a full asymptotic expansion rather than merely a qualitative limit (Brunault et al., 2022).

Taken together, these results show that “multivariable Boyd–Lawton theorem” now designates a robust approximation principle: torus averages of singular logarithmic-type functions are stable under increasingly generic monomial substitutions, whether the target is ordinary Mahler measure, its higher and generalized variants, or closely related dynamical and zero-counting analogues (Aitken et al., 8 Aug 2025).