Auxiliary function

In mathematics, particularly in transcendental number theory, an auxiliary function is a specially constructed mathematical object—often a polynomial, rational function, or more complex analytic expression—designed to facilitate proofs of transcendence, irrationality, or algebraic independence of numbers such as e, \pi, or values of the gamma function. These functions are engineered to exhibit controlled behavior, such as having small values or specific zeros at algebraic points of interest, which enables the application of inequalities like Liouville's theorem or the Schwarz lemma to derive contradictions or precise bounds that confirm the transcendental nature of the target quantities.^[1] The development of auxiliary functions traces back to the mid-19th century, beginning with Joseph Liouville's 1844 construction of polynomials to demonstrate the transcendence of certain infinite series, such as \sum 10^{-n!}, marking the first explicit proof of a transcendental number.^[1] Charles Hermite advanced this in 1873 by introducing Padé approximants as auxiliary functions to prove the transcendence of e, using forms like B(z)e^z - A(z) where A(z) and B(z) are polynomials tailored to minimize discrepancies at integer points.^[1] Subsequent refinements by mathematicians including Axel Thue, Carl Ludwig Siegel, Aleksandr Gelfond, and Theodor Schneider in the early 20th century incorporated tools like the Thue–Siegel lemma (based on Dirichlet's box principle) to handle multivariable cases, enabling proofs for numbers like e^\pi and \pi.^[1] By the mid-20th century, Kurt Mahler introduced linear algebra techniques for constructing auxiliary functions to solve functional equations, while later innovations, such as Michel Laurent's 1991 use of interpolation determinants, eliminated reliance on Dirichlet's principle for more efficient constructions.^[1] These functions have been instrumental in addressing major open problems, including Hilbert's seventh problem on the transcendence of a^b for algebraic a \neq 0,1 and irrational algebraic b, and continue to underpin modern results in algebraic independence, such as for values of the gamma function at rational arguments.^[1]

Fundamentals

Definition

In transcendental number theory, which concerns numbers that are not roots of any non-zero polynomial equation with rational coefficients, auxiliary functions serve as essential tools for proving irrationality or transcendence. These functions address the fundamental challenge of distinguishing algebraic numbers—those satisfying such polynomial equations—from transcendentals by constructing approximations that algebraic numbers cannot achieve due to limitations imposed by their degree and height. Specifically, transcendence proofs often rely on showing that certain numbers can be approximated by rationals or algebraics to an extent that violates known bounds for algebraic numbers, such as Roth's theorem or Liouville's inequality extensions.^[2]^[3] An auxiliary function is a specially constructed analytic or algebraic function, typically a polynomial, rational approximant, or integral expression, engineered to exhibit particular boundedness properties, such as taking exceptionally small values at rational or algebraic points of interest while remaining non-zero overall. The core purpose is to derive a contradiction: if the number in question were algebraic, the function's smallness at those points would imply it vanishes identically or equals zero, which it does not, thus establishing transcendence. These functions are derived using techniques like the pigeonhole principle, linear algebra (e.g., Thue-Siegel lemma), or interpolation to ensure the desired estimates.^[2]^[3] Common general forms include polynomial sums like \sum_{k=0}^m a_k x^k / k! where x is algebraic and coefficients a_k are chosen rationally. Key properties of auxiliary functions encompass their non-integral evaluation at algebraic points (ensuring they are not integers or algebraic integers in the assumed case), strict positivity in many analytic constructions to avoid sign changes, and established lower bounds on their minima or norms, which prevent trivial zero behavior and support the contradiction. These properties are rigorously controlled through estimates on derivatives, growth rates, or zero multiplicities at specified points.^[2]^[3]

Historical Development

The origins of auxiliary functions in mathematical proofs trace back to Joseph Liouville's pioneering work, with his approximation theorem in 1844 laying groundwork through Diophantine approximation to show that certain real numbers are not algebraic. In his seminal 1851 memoir "Sur des classes très-étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationnelles algébriques," published in the Journal de Mathématiques Pures et Appliquées, Liouville employed these functions to construct numbers with exceptionally good rational approximations, thereby proving their transcendence and laying the groundwork for the field.^[4] Advancements in the 19th century built on this foundation, with Joseph Fourier's 1815 argument for the irrationality of e—published posthumously—utilizing infinite series to generate sharp rational approximations, serving as an early precursor to auxiliary function techniques in irrationality and transcendence proofs.^[2] Charles Hermite advanced the method significantly in 1873 by proving the transcendence of e through the construction of an explicit auxiliary function involving integrals of exponential terms, which demonstrated that e satisfies no algebraic equation with rational coefficients; this memoir in the Comptes Rendus de l'Académie des Sciences marked the first such proof for a specific transcendental number.^[4] In the 20th century, Ferdinand von Lindemann extended Hermite's approach in 1882 to prove the transcendence of π by showing that e^{iπ} = -1 implies π's non-algebraicity via auxiliary functions approximating exponential integrals.^[5] This culminated in the 1934 Gelfond-Schneider theorem, independently proved by Aleksandr Gelfond and Theodor Schneider, which established the transcendence of numbers of the form α^β for algebraic α ≠ 0,1 and irrational algebraic β, using auxiliary polynomials in entire functions like e^z and e^{βz}. The evolution of auxiliary functions shifted from these explicit, integral-based constructions in the 19th century to more abstract formulations by the mid-20th century, incorporating the pigeonhole principle for Diophantine approximation bounds and interpolation methods for approximating transcendental functions with algebraic ones.^[2]

Explicit Constructions

Liouville's Transcendence Criterion

Liouville's transcendence criterion, published in 1844, establishes a sufficient condition for a real number \alpha to be transcendental based on the quality of its rational approximations. If \alpha is algebraic of degree n, then there exists a positive constant c (depending on \alpha) such that |\alpha - p/q| > c / q^n for all integers p, q with q > 0. Thus, if there are infinitely many rationals p/q satisfying |\alpha - p/q| < 1/q^\kappa for some \kappa > n, then \alpha cannot be algebraic and must be transcendental.^[2] The proof relies on constructing an auxiliary function from the minimal polynomial of \alpha. Assume \alpha is algebraic of degree n with irreducible minimal polynomial f(x) \in \mathbb{Z} of degree n, so f(\alpha) = 0. For a rational p/q in lowest terms, q^n f(p/q) is a non-zero integer because f is irreducible over \mathbb{Q}, implying |f(p/q)| \geq 1/q^n. This polynomial f serves as the auxiliary function.^[2] To derive the approximation bound, apply the mean value theorem: |f(\alpha) - f(p/q)| \leq \sup |f'(x)| \cdot |\alpha - p/q| over the interval between \alpha and p/q, where the supremum is bounded by some constant M depending on f and the interval. Since f(\alpha) = 0, it follows that |f(p/q)| \leq M |\alpha - p/q|, so |\alpha - p/q| \geq 1/(M q^n). If approximations better than this bound exist infinitely often (i.e., with exponent \kappa > n), the assumption that \alpha is algebraic leads to a contradiction. The key inequality is thus |f(p/q)| > c / q^\mu for \mu = n and some c > 0, which cannot hold if |\alpha - p/q| is sufficiently small.^[6] A seminal application is to Liouville's constant \alpha = \sum_{k=1}^\infty 10^{-k!}, the first explicit transcendental number constructed in 1844. Consider the partial sum up to m, p/q = \sum_{k=1}^m 10^{-k!} with q = 10^{m!}, so p, q are integers. The tail satisfies |\alpha - p/q| = \sum_{k=m+1}^\infty 10^{-k!} < 10^{-(m+1)!} / (1 - 10^{-(m+1)!}) < 2 \cdot 10^{-(m+1)!} for m \geq 2. Since (m+1)! = (m+1) \cdot m!, this is |\alpha - p/q| < 2 / q^{m+1}. For fixed degree n, choose m > n; then \kappa = m+1 > n, yielding infinitely many such approximations as m increases, so \alpha is transcendental by the criterion using the auxiliary minimal polynomial.^[2]

Fourier's Proof of the Irrationality of e

Fourier outlined a proof of the irrationality of e in a manuscript dated around 1815, which was communicated by Louis Poinsot and published the same year in Janot de Stainville's Mélanges d'analyse algébrique et de géométrie.^[7]^[8] This proof represents the first rigorous demonstration that e cannot be expressed as a ratio of integers, predating more advanced results on its transcendence.^[7] The proof centers on the Taylor series expansion of e:

e = \sum_{k=0}^{\infty} \frac{1}{k!}.

The auxiliary function is the remainder term after truncating the series at n terms,

R_n = e - \sum_{k=0}^{n} \frac{1}{k!} = \sum_{k=n+1}^{\infty} \frac{1}{k!},

which captures the tail of the exponential series and satisfies $0 < R_n < \frac{1}{n \cdot n!}.^[7] This remainder can equivalently be expressed in integral form to aid in bounding its value,

R_n = \int_{0}^{1} \frac{(1-t)^n}{n!} e^{t} \, dt,

derived from the integral form of the Taylor remainder theorem applied to e^x at x=1.^[9] The integral representation highlights the positive nature and upper bound of the tail, with

R_n > \int_{0}^{1} \frac{(1-t)^n}{n!} \, dt = \frac{1}{(n+1)!}

and

R_n < \frac{e}{(n+1)!},

ensuring $0 < n! R_n < \frac{e}{n+1} < 1 for sufficiently large n.^[9] To establish irrationality, assume for contradiction that e = a/b where a and b are positive integers with \gcd(a,b)=1. Set n = b. Then b! e = a \cdot (b-1)!, which is an integer.^[7] Alternatively, expanding the series gives

b! e = \sum_{k=0}^{b} \frac{b!}{k!} + b! R_b,

where \sum_{k=0}^{b} \frac{b!}{k!} is an integer because k! divides b! for k \leq b.^[7] Thus, b! R_b must also be an integer. However, the bounds on the remainder imply

\frac{1}{b+1} < b! R_b < \frac{e}{b+1} < 1

for b \geq 3, so $0 < b! R_b < 1, contradicting the assumption that it is a nonzero integer.^[7]^[9] Therefore, e is irrational. A key aspect of the proof involves scaling the remainder via integrals related to the gamma function, where the full integral satisfies

\int_{0}^{\infty} e^{-t} \frac{t^n}{n!} \, dt = 1,

providing context for the tail's magnitude when truncated and scaled appropriately to yield the strict inequality $0 < b! R_b < 1.^[7] This approach using the series tail as an auxiliary function marks the earliest known rigorous argument for e's irrationality, relying on the exponential series without invoking continued fractions or other methods.^[8]

Hermite's Proofs Involving e

In 1873, Charles Hermite extended earlier work on the irrationality of e, such as Fourier's 1815 proof using integral remainders of the exponential series, to establish the irrationality of e^r for any nonzero rational r and the transcendence of e itself. His contributions appeared in a series of notes in the Comptes rendus hebdomadaires des séances de l'Académie des sciences and were consolidated in a memoir published in Journal de mathématiques pures et appliquées (Liouville's Journal). These proofs relied on explicit constructions of auxiliary functions via Padé approximants and integrals, which allowed Hermite to derive contradictions from assumptions of algebraicity by bounding the functions' values at integers while preserving integrality properties.^[10]^[2] For the irrationality of e^r where r is a nonzero rational, Hermite constructed auxiliary functions as rational approximations to e^z, specifically polynomials A(z) and B(z) with integer coefficients such that the auxiliary R(z) = B(z) e^z - A(z) has a zero of high multiplicity at z=0. Evaluating at z=r yields B(r) e^r - A(r) = R(r), where $0 < |R(r)| < 1 for large multiplicity after appropriate scaling (e.g., multiplying by a factorial denominator to make A(r), B(r) integers), contradicting the assumption that e^r = a/b rational, as it would imply R(r) = 0. This approach generalized Fourier's integral-based remainder estimates for the exponential function.^[11]^[12] To prove the transcendence of e, Hermite extended these constructions using auxiliary polynomials f(t) with high-order zeros at integers t=0,1,\dots,m, such as f_r(t) = t^{r-1} (t-1)^r \cdots (t-m)^r, and integrals I_k = \int_0^k e^{-t} f(t) \, dt for k=0,1,\dots,m. Under the assumption that e satisfies a polynomial equation \sum_{j=0}^d a_j e^j = 0 with a_j \in \mathbb{Z}, a_d \neq 0, a linear combination \Phi = \sum_{j=0}^d a_j e^j I_j (adjusted via Hermite's identity relating integrals and polynomials) is a nonzero algebraic integer. However, bounds show $0 < |\Phi| < 1 for sufficiently large r > d, using factorial decay in the integrals, leading to a contradiction since no such small nonzero algebraic integer exists. This integral satisfies properties ensuring no exact cancellation, ruling out algebraic relations.^[11]^[13]^[2] These methods marked a pivotal advance, introducing systematic use of auxiliary integrals and Padé approximants for Diophantine approximations in transcendental number theory.^[14]^[15]

Pigeonhole Principle Applications

Auxiliary Polynomial Theorem

In Diophantine approximation, auxiliary polynomials are constructed using the pigeonhole principle to obtain non-zero polynomials with integer coefficients that take exceptionally small values at a given algebraic number \alpha of degree d over \mathbb{Q} and bounded height. These polynomials enable effective lower bounds on how well \alpha can be approximated by rationals.^[16] The construction often begins by applying the pigeonhole principle in the unit torus to vectors of evaluations (P(0), P(1), \dots, P(m)) modulo 1 in [0,1]^{m+1}, where m is on the order of d. By considering a large collection of polynomials with bounded integer coefficients, their evaluation vectors fill the torus. With more vectors than subintervals of a suitable grid, the principle guarantees two distinct polynomials whose difference P (non-zero) has evaluation vector components differing by less than the grid spacing, yielding small fractional parts and thus small |P(k)| for integers k = 0 to m. This framework, foundational to the Thue-Siegel method, extends to the conjugates of \alpha using tools like Siegel's lemma—a pigeonhole-based result on small solutions to linear systems—to achieve simultaneous smallness |P(\beta_i)| < \varepsilon for each Galois conjugate \beta_i of \alpha (i = 1, \dots, d), where \varepsilon is controlled by the grid size $1/N and N exceeds the number of candidates. This uniform control ensures the polynomial is non-trivial while providing quantitative estimates for contradiction arguments in approximation theorems.^[16] The degree n and height of such P are balanced to make |P(\alpha)| small relative to the height H(P), typically achieving |P(\alpha)| \ll H(P)^{-\mu} for some \mu > 1 depending on d, which underpins irrationality measures for algebraic numbers. The height Q of \alpha, reflecting the coefficients of its minimal polynomial, influences the scale of the approximation.^[16] This approach serves as a precursor to advanced results like Roth's theorem on the irrationality measure of algebraic numbers, by enabling controlled constructions of polynomials small near algebraic points.^[16]

Lang's Theorem on Diophantine Approximation

The Schneider-Lang theorem on Diophantine approximation, a refinement by Serge Lang of Theodor Schneider's work in the 1960s, provides a powerful criterion for limiting the algebraic values taken by entire functions of finite order, with significant implications for transcendental number theory.^[17] Specifically, consider meromorphic functions f_1, \dots, f_m in \mathbb{C} of finite order, where f_1 and f_2 are algebraically independent over \mathbb{Q}(z), and the derivatives satisfy f_j'(w) \in K(f_1(w), \dots, f_m(w)) for a number field K. Then, the set S = \{ w \in \mathbb{C} \mid w is not a pole of any f_j, and f_j(w) \in K for all j = 1, \dots, m \} is finite.^[3] In the case of a single entire function f(z) of finite order \rho, this implies that there are only finitely many rationals p/q (in lowest terms) such that |f(p/q)| < 1/|q|^\kappa for any \kappa > \rho, unless f is a special function satisfying an algebraic differential equation over \mathbb{Q}(z).^[18] The proof relies on constructing auxiliary functions via the pigeonhole principle to exploit the growth properties of entire functions. One key step involves Dirichlet's pigeonhole principle applied to lattice points in the space of integer linear combinations of basis functions derived from the f_j. This allows selection of non-trivial integer coefficients b_1, \dots, b_n such that the auxiliary function g(z) = \sum b_i \prod f_j(z)^{a_{ij}} (or a similar form) vanishes at many points in S \cap D(0, R), where D(0, R) is a disk of radius R.^[17] For applications involving approximations, auxiliary functions of the form \exp(g(z)), where g(z) is a polynomial with algebraic coefficients, are used to bound distances like |f(\alpha) - \beta| for algebraic \alpha, \beta \in K, leveraging the finite order to control growth outside the disk.^[19] In the proof, after constructing g(z) to have at least N zeros within a smaller disk |z| \leq r < R, analytic estimates such as the Schwarz lemma provide an upper bound: |g(0)| \leq \left( \frac{3r}{R} \right)^N \max_{|z|=R} |g(z)|.^[17] A lower bound for |g(0)| is then obtained via Diophantine approximation on the coefficients, ensuring that if S were infinite, the growth would contradict the finite order unless the functions are dependent. This builds on the auxiliary polynomial theorem by extending algebraic pigeonhole arguments to analytic settings.^[18] A primary application lies in Baker's method for lower bounds on linear forms in logarithms, where Lang's framework refines estimates for forms \Lambda = b_0 + b_1 \log \alpha_1 + \dots + b_n \log \alpha_n with algebraic \alpha_i and integer b_i. Using auxiliary polynomials in the exponents, the method yields |\Lambda| > \exp(-C (\log H)^\tau), where H is the (exponential) height of the \alpha_i and b_i, and C, \tau > 0 depend on the degree and n. This quantitative result, pivotal for solving Diophantine equations, stems directly from the Diophantine control imposed by the theorem on approximations near algebraic points.^[20]

Interpolation Methods

General Interpolation Determinants

In transcendence theory, auxiliary functions are often constructed using interpolation determinants, which are scalars derived from matrices whose entries involve evaluations of a function and its derivatives at specified points. Specifically, for an analytic function f and distinct points a_1, \dots, a_n, the interpolation matrix M has entries M_{ij} = f^{(i-1)}(a_j) / (i-1)! for i, j = 1, \dots, n, and the auxiliary is the determinant \det(M). This construction generalizes the Vandermonde determinant from polynomial interpolation to analytic functions, providing a measure of linear independence (or near-dependence) among the vectors of function values and scaled derivatives at the points a_j. A small |\det(M)| indicates that f is well-approximated by a polynomial of degree less than n at these points, which is key for deriving upper bounds in transcendence arguments.^[21] Upper bounds on the magnitude of \det(M) are obtained using Hadamard's inequality, which limits the growth based on the row norms, often combined with analytic estimates like the maximum modulus principle to bound derivatives within a disk of convergence. For instance, if f is entire and bounded on a disk of radius R, the determinant satisfies |\det(M)| \leq n^{n/2} \prod_{j=1}^n \max_{0 \leq k < n} |f^{(k)}(a_j)/k!|. Lower bounds in generic cases ensure non-vanishing, with estimates such as |\det(M)| > \exp(-n^2 \log n) for suitably chosen points. These properties enable tight control over approximation quality, crucial for contradictions in transcendence proofs.^[22] The method traces its historical roots to the late 19th century, stemming from Weierstrass's work on entire functions and interpolation techniques to bound zeros. It was refined in the 20th century, notably by Schneider, and formalized by M. Laurent in 1991 through explicit determinant constructions for exponential polynomials that avoid the pigeonhole principle. A foundational example is the Vandermonde determinant \Delta = \det((x_j^{i-1})_{1 \leq i,j \leq n}), which for points x_j related to exponential bases (e.g., x_j \approx e^{\alpha_j}) and |x_j| \leq 1 admits lower bounds such as |\Delta| > \exp(-n^2 \log n) from product formulas and potential theory, ensuring non-vanishing in generic configurations.^[1] These interpolation determinants bridge classical interpolation with analytic estimates, facilitating global transcendence results from local approximations. A prominent application is in proofs of the Hermite–Lindemann theorem, where they construct linear forms in exponential values at algebraic points to establish transcendence.^[21]

Proof of the Hermite–Lindemann Theorem

The Hermite–Lindemann theorem asserts that if \alpha is a non-zero algebraic number, then e^{\alpha} is transcendental. A key consequence is the transcendence of \pi, since e^{i\pi} = -1 and i\pi is algebraic, so assuming \pi algebraic would imply e^{i\pi} algebraic, contradicting the theorem.^[23]^[24] Lindemann's 1882 proof extends Hermite's 1873 demonstration of the transcendence of e by generalizing the auxiliary function approach to algebraic exponents.^[23]^[25] To establish transcendence, assume for contradiction that e^{\alpha} is algebraic, where \alpha \neq 0 is algebraic of degree d. Consider the field extension \mathbb{Q}(\alpha, e^{\alpha}) of degree at most d^2, and let \alpha_1 = \alpha, \dots, \alpha_m be the conjugates of \alpha under the Galois group, with m \leq d. The assumption implies e^{\alpha_j} are algebraic for all j.^[25] The core of the proof relies on constructing an auxiliary function via interpolation determinants to capture linear relations among the e^{\alpha_j}. Specifically, the coefficients b_1, \dots, b_m are defined using determinants of matrices that enforce interpolation conditions on the exponentials at integer points. This yields the auxiliary function \phi(z) = \sum_{j=1}^m b_j e^{\alpha_j z}, which satisfies \phi(k) = 0 for integers k = 1, \dots, n (with n large) under the linear dependence assumption derived from the algebraic hypothesis.^[23]^[25] Under the assumption, \phi(z) is non-zero (as the e^{\alpha_j z} are linearly independent over \overline{\mathbb{Q}} in generic settings), and \phi(0) = \sum b_j is a non-zero element of the number field with bounded denominator (from the field properties and integer coefficients in the determinants), implying |\phi(0)| \geq c > 0 for some positive constant c independent of n. However, estimates on \phi(z) using the multiple zeros at k=1,\dots,n, combined with Hadamard's inequality on the coefficient determinants and analytic bounds on the exponentials, yield $0 < |\phi(0)| < \exp(-c n^2) for large n and suitable c > 0. This smallness arises from the interpolation conditions suppressing the values near the origin. The contradiction between the lower bound from algebraicity and the exponential upper bound proves the assumption false.^[23]^[25] The auxiliary \phi(z) satisfies a linear differential equation of order m (the number of distinct \alpha_j), as it is a linear combination of solutions to first-order DEs y' = \alpha_j y. Applying the Phragmén–Lindelöf principle to this entire function on suitable contours ensures \phi(z) does not vanish identically and reinforces the lower bound at z=0. This differential structure, combined with interpolation properties, distinguishes the proof from earlier methods and solidifies the transcendence result.^[23]^[25]