Charles Hermite

How to use this course

Charles Hermite was the central figure of French mathematics in the second half of the nineteenth century. His results are scattered across the curriculum: Hermitian matrices in linear algebra and quantum mechanics, Hermite polynomials in probability and physics, Hermite interpolation in numerical analysis, Hermite’s constant in the geometry of numbers, and the first proof that a classical constant is transcendental. Behind them is one habit of mind. Hermite moved freely between arithmetic and analysis, and used continuous variables and analytic functions to prove facts about integers.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern language, proof sketches and exercises with folded solutions. The order is roughly that of his work: quadratic forms (1847–1855), elliptic and modular functions (1843–1858), transcendence and approximation (1873), and orthogonal polynomials and interpolation (1864–1878).

Timeline

Dieuze, Nancy, Paris, 1822–1842

Charles Hermite was born on 24 December 1822 in Dieuze, in Lorraine, the sixth of seven children of Ferdinand Hermite and Madeleine Lallemand. His father had trained as an engineer and worked at a salt mine near the town before going into the drapery trade; he also had an artist’s interests. When Charles was about seven the family moved to Nancy. He was born with a deformity of the right foot that made walking difficult all his life.

He went to school at the Collège de Nancy and then in Paris, first at the Collège Henri IV and from 1840 at Louis-le-Grand. There his mathematics teacher was Louis Richard, who had taught Évariste Galois about a decade earlier and recognized the same kind of talent. Hermite did not care for the routine of examination preparation. He read Leonhard Euler, Joseph-Louis Lagrange and Carl Friedrich Gauss instead.

Before entering any grande école he had published two papers in the Nouvelles Annales de Mathématiques. One was a simplified argument that the general equation of the fifth degree cannot be solved by radicals, the result of Niels Henrik Abel and Paolo Ruffini. Sixteen years later he would show how the quintic can be solved by other means (Lesson 07).

The Polytechnique and the letters to Jacobi, 1842–1847

Tutored by Eugène Catalan, Hermite passed the entrance examination of the École Polytechnique in 1842, ranked 68th. After his first year the school refused to let him continue because of his disability. Friends had the decision reversed, but only under conditions he found unacceptable, and he left without graduating.

He spent the next years working on his own. In January 1843, aged twenty, he wrote to Carl Gustav Jacob Jacobi in Königsberg. Abel had shown how to divide the argument of an elliptic function, expressing \(f(u/n)\) through algebraic equations in \(f(u)\). Hermite extended this to the abelian functions of hyperelliptic integrals that Jacobi was then developing. Jacobi replied with encouragement. A further series of letters, on the arithmetic of quadratic forms, followed and appeared in Crelle’s Journal in 1850.

The young man was now known in Berlin and Königsberg before he had a French degree. He frequented the home of the mathematician Joseph Bertrand, and later married Bertrand’s sister Louise. In 1847 he finally took the baccalauréat and licence.

Examiner and academician, 1848–1862

In 1848 Hermite was appointed répétiteur, a tutor, and admissions examiner at the École Polytechnique, the school that had pushed him out. He also taught for a time at the Collège de France as a substitute. These were modest posts, and for more than a decade they were all he held. The research was not modest. Between 1847 and 1858 he published his arithmetic theory of quadratic forms, the Hermitian forms, his reciprocity law in invariant theory, and the solution of the quintic.

In 1849 he submitted a memoir on doubly periodic functions that applied Cauchy’s residue calculus. Cauchy and Charles Sturm reported on it favourably, but it was not published at the time, apparently because of a priority dispute with Joseph Liouville, who was developing the theory of elliptic functions from his own theorem on bounded entire functions.

In 1856 Hermite fell seriously ill with smallpox. Cauchy, a devout Catholic, supported him through the illness, and Hermite, until then indifferent to religion, became a convinced Catholic. On 14 July 1856 the Académie des sciences elected him to the seat left by Jacques Binet. He was 33. In 1862 Louis Pasteur, then director of scientific studies at the École Normale Supérieure, had a post of maître de conférences created for him there, and from 1863 he was also a permanent examiner at the Polytechnique.

Quadratic forms and Hermite’s constant

For the general reader. How close can an integer point come to the origin when distance is measured by a quadratic formula? Gauss had answered for two variables. In his letters to Jacobi, Hermite answered for any number of variables, and used the answer to prove finiteness theorems in number theory. This is the beginning of what Hermann Minkowski later called the geometry of numbers.

Let \(f(x)=x^{T}Ax\) be a positive definite quadratic form in \(n\) variables with real symmetric matrix \(A\), and let \(\det f=\det A\). Its arithmetic minimum is \(m(f)=\min\{f(x): x\in\mathbb Z^n\setminus\{0\}\}\).

Proof sketch. Induction on \(n\); for \(n=1\) it is an equality. Choose \(v\in\mathbb Z^n\) with \(f(v)=m(f)\). Then \(v\) is primitive, so it extends to a basis of \(\mathbb Z^n\), and after an integral change of variables \(f(x)=m\,(x_1+\ell(x'))^2+g(x')\) with \(x'=(x_2,\dots,x_n)\), \(\ell\) linear and \(g\) a positive form in \(n-1\) variables with \(\det g=\det f/m\). By induction there is \(y'\ne0\) with \(g(y')\le(4/3)^{(n-2)/2}(\det g)^{1/(n-1)}\). Choose \(y_1\in\mathbb Z\) with \(|y_1+\ell(y')|\le\frac12\). Then

Raising to the power \(n-1\) and rearranging gives \(m^n\le(4/3)^{n(n-1)/2}\det f\), which is the theorem.

Hermite’s constant

The best possible constant \(\gamma_n=\sup_f m(f)/(\det f)^{1/n}\) is called Hermite’s constant. Hermite’s theorem says \(\gamma_n\le(4/3)^{(n-1)/2}\). Minkowski’s convex body theorem of 1889 gave a bound linear in \(n\), much better for large \(n\). The exact value is known only for \(n\le8\) and \(n=24\): \(\gamma_2^2=4/3\), \(\gamma_3^3=2\), \(\gamma_4^4=4\), up to \(\gamma_8=2\) (the \(E_8\) lattice) and \(\gamma_{24}=4\) (the Leech lattice, proved by Cohn and Kumar in 2009). Finding \(\gamma_n\) is the lattice sphere-packing problem.

Finiteness

The same reduction shows that every positive form with integer coefficients is equivalent under \(\mathrm{GL}_n(\mathbb Z)\) to one whose coefficients are bounded in terms of the determinant. Hence there are finitely many classes of integral forms of given determinant. Hermite applied such arguments to algebraic number fields and proved that only finitely many number fields have a given discriminant.

Hermitian forms and reciprocity

For the general reader. A symmetric real matrix has real eigenvalues. Hermite saw that the right analogue for complex numbers is not a symmetric matrix but one equal to its own conjugate transpose. These Hermitian matrices are now the observables of quantum mechanics. In the same years he found a surprising symmetry in the invariant theory of Arthur Cayley and James Joseph Sylvester.

A Hermitian form in \(n\) complex variables is \(h(z)=\sum_{i,j}a_{ij}z_i\bar z_j\) with \(a_{ji}=\bar a_{ij}\), that is \(h(z)=z^{*}Az\) with \(A^{*}=A\). Its values are real. Hermite introduced such forms in 1854–55, in arithmetic work on forms with complex integer coefficients, and extended his reduction theory of Lesson 04 to them.

Proof. If \(Av=\lambda v\) with \(v\ne0\), then \(\lambda\,v^{*}v=v^{*}Av=(v^{*}Av)^{*}=\bar\lambda\,v^{*}v\), and \(v^{*}v>0\), so \(\lambda=\bar\lambda\). If \(v\) is an eigenvector, the orthogonal complement \(v^\perp\) is invariant under \(A\), since \(w\perp v\) implies \(\langle Aw,v\rangle=\langle w,Av\rangle=\bar\lambda\langle w,v\rangle=0\). Induction on dimension gives an orthonormal eigenbasis. Cauchy had proved the real symmetric case in 1829; Hermite’s remark extended it.

The reciprocity law

Recall from the Cayley course that the invariants of degree \(d\) of a binary form of order \(n\) are counted by \(p(nd/2;d,n)-p(nd/2-1;d,n)\), where \(p(w;d,n)\) counts partitions of \(w\) into at most \(d\) parts, each at most \(n\).

Proof via partitions. A partition into at most \(d\) parts each at most \(n\) is a Young diagram inside a \(d\times n\) box. Transposing the diagram gives a partition into at most \(n\) parts each at most \(d\), of the same size. So \(p(w;d,n)=p(w;n,d)\), and the counting formula is symmetric in \(d\) and \(n\). The same argument matches the weight multiplicities of the two representations, which determine them.

Elliptic functions and theta

For the general reader. The sine function repeats itself every \(2\pi\). Elliptic functions repeat in two independent complex directions, so that their values are determined by a parallelogram. Hermite’s first research, and much of his teaching, concerned these functions and the theta series from which Jacobi built them.

Let \(\Lambda=\mathbb Z\omega_1+\mathbb Z\omega_2\) be a lattice in \(\mathbb C\). An elliptic function is a meromorphic \(f\) with \(f(z+\omega)=f(z)\) for all \(\omega\in\Lambda\).

Proof. (1) A holomorphic elliptic function is bounded, since it is continuous on a compact parallelogram, so it is constant by Liouville’s theorem. (2) Integrate \(f\) around the boundary of a parallelogram with no poles on it. Opposite sides cancel by periodicity, so the integral, and hence the sum of residues, is 0. (3) Apply (2) to \(f'/(f-c)\), which is elliptic: the number of zeros of \(f-c\) minus the number of poles is 0. If the order were 1, \(f\) would have a single simple pole with non-zero residue, contradicting (2).

Hermite’s 1848 work showed that every elliptic function is a quotient of entire functions that are periodic in one direction and quasi-periodic in the other; theta functions are the basic examples. He later gave the decomposition of an elliptic function into simple elements, analogous to partial fractions, that bears his name.

Theta functions

For \(\operatorname{Im}\tau>0\) let

It satisfies \(\vartheta(z+1,\tau)=\vartheta(z,\tau)\) and \(\vartheta(z+\tau,\tau)=e^{-\pi i\tau-2\pi i z}\vartheta(z,\tau)\). Under the modular group it transforms by

In 1858 Hermite determined the eighth roots of unity that appear in the transformation of the four Jacobi theta functions under a general element of \(\mathrm{SL}_2(\mathbb Z)\). This was the technical heart of his solution of the quintic.

Solving the quintic

For the general reader. Abel and Galois showed that the general equation of degree five cannot be solved using only arithmetic and root extraction. That leaves open whether some other, richer function can do the job, in the way that the cosine solves the cubic. In 1858 Hermite showed that elliptic modular functions solve the quintic. Leopold Kronecker and Francesco Brioschi reached the same goal by related routes the same year.

The model: cubics and cosines

The cubic \(t^3-3t-2a=0\) with \(|a|\le1\) is solved by \(t=2\cos\big(\tfrac13\arccos a\big)\). The function \(\cos\) has the property that \(\cos3\theta\) is a polynomial in \(\cos\theta\); dividing the argument by 3 solves a cubic. Hermite’s solution replaces trigonometric functions by modular functions and division of the argument by transformation of order 5.

Reduction

Hermite’s solution

Let \(k\) be the modulus of an elliptic function with \(k^2=\lambda(\tau)\), and set \(u=k^{1/4}\), a modular function of \(\tau\). Jacobi’s modular equation of order 5 relates \(u=u(\tau)\) and \(v=u(5\tau)\):

For fixed \(u\) this has six roots \(v_0,\dots,v_5\), corresponding to the six transformations of order 5. In modern terms the monodromy group of this sextic is \(\mathrm{PSL}_2(\mathbb F_5)\cong A_5\). Galois had stated, in his last letter of 1832, that the modular equation of degree 6 can be lowered to degree 5. Hermite wrote down explicit products of differences of the \(v_j\), five in number, that are permuted among themselves by the monodromy, and showed that they satisfy a quintic of Bring–Jerrard form \(y^5-y-a=0\) after rescaling, with \(a\) an explicit algebraic function of the modulus \(k\). Given \(a\), one solves an algebraic equation for \(k\), then finds \(\tau\) by elliptic integrals, and the roots are values of modular functions at \(\tau\) and its transforms.

Felix Klein’s Lectures on the Icosahedron (1884) recast the whole construction through the symmetries of the icosahedron.

The transcendence of e

For the general reader. A number is algebraic if it solves a polynomial equation with integer coefficients, and transcendental otherwise. Joseph Liouville constructed the first transcendental numbers in 1844, but they were made for the purpose. In 1873 Hermite proved that \(e\), the base of natural logarithms, is transcendental. Nine years later Lindemann used the same method to prove that \(\pi\) is transcendental, which settled the ancient problem of squaring the circle.

Hermite’s idea

Hermite found polynomials \(P_0,\dots,P_n\) such that \(P_0(x)e^{kx}-P_k(x)\) is very small near \(x=0\) for every \(k\), and evaluated at \(x=1\). This gives rational numbers \(P_k(1)/P_0(1)\) that approximate \(e,e^2,\dots,e^n\) simultaneously, and too well for a linear relation to survive. These are now called Hermite–Padé approximants, after Hermite’s student Henri Padé. The streamlined proof below follows David Hilbert (1893) and Adolf Hurwitz.

Proof

For a polynomial \(f\) of degree \(m\) put \(F=f+f'+\cdots+f^{(m)}\). Integration by parts gives

Suppose \(\sum_{k=0}^n a_ke^k=0\) with \(a_0\ne0\) (divide by a power of \(e\) if needed). Multiply the identity by \(a_k\) and sum:

Take a large prime \(p>\max(n,|a_0|)\) and \(f(x)=\frac{1}{(p-1)!}x^{p-1}(x-1)^p\cdots(x-n)^p\). Every derivative \(f^{(j)}(k)\) for \(k=1,\dots,n\) is an integer divisible by \(p\), and so is \(f^{(j)}(0)\) for \(j\ne p-1\), while \(f^{(p-1)}(0)=(\pm n!)^p\), which is not divisible by \(p\). So the left side is an integer congruent to \(-a_0(\pm n!)^p\not\equiv0\pmod p\), hence of absolute value at least 1. On \([0,n]\), \(|f(x)|\le n^{(n+1)p}/(p-1)!\), so the right side is at most \(C\cdot c^p/(p-1)!\) for constants \(C,c\), which tends to 0. For large \(p\) this is a contradiction.

Hermite himself doubted that the method would extend to \(\pi\). Lindemann proved in 1882 that \(e^\alpha\) is transcendental for every non-zero algebraic \(\alpha\); since \(e^{i\pi}=-1\), \(i\pi\) and hence \(\pi\) is transcendental. Karl Weierstrass gave the general Lindemann–Weierstrass theorem in 1885.

Hermite polynomials and interpolation

For the general reader. The bell-shaped Gaussian curve has a family of companion polynomials that let any reasonable function be expanded in Gaussian “harmonics”. Pierre-Simon Laplace and Pafnuty Chebyshev met these polynomials first; Hermite’s 1864 paper studied them systematically, including in several variables. Seventy years later they appeared as the wave functions of the quantum harmonic oscillator, drawn at the top of this page.

Definition and properties

The (physicists’) Hermite polynomials are

Proofs. (1) By Taylor’s theorem, \(e^{-(x-t)^2}=\sum_n\frac{t^n}{n!}\frac{d^n}{dt^n}e^{-(x-t)^2}\big|_{t=0}=\sum_n\frac{t^n}{n!}(-1)^n\frac{d^n}{dx^n}e^{-x^2}\); multiply by \(e^{x^2}\). (2) Differentiate the generating function \(G=e^{2xt-t^2}\) in \(t\) and in \(x\): \(\partial_tG=(2x-2t)G\) and \(\partial_xG=2tG\), and compare coefficients. (3) Combine the two relations in (2). (4) For \(m<n\), integrate by parts \(n\) times: \(\int H_m(-1)^n\frac{d^n}{dx^n}e^{-x^2}dx=\int H_m^{(n)}e^{-x^2}dx=0\). For \(m=n\), \(H_n^{(n)}=2^nn!\) and \(\int e^{-x^2}=\sqrt\pi\).

The quantum oscillator

The Schrödinger equation \(-\tfrac12\psi''+\tfrac12x^2\psi=E\psi\) (units with \(\hbar=m=\omega=1\)) has square-integrable solutions exactly when \(E=n+\tfrac12\), and then \(\psi_n(x)=c_nH_n(x)e^{-x^2/2}\). Substituting \(\psi=He^{-x^2/2}\) turns the equation into \(H''-2xH'+(2E-1)H=0\), Hermite’s equation with \(2n=2E-1\).

Hermite interpolation (1878)

Uniqueness: the difference of two interpolants has \(N\) zeros counted with multiplicity and degree below \(N\). Existence then follows because the linear map from polynomials of degree below \(N\) to the \(N\) data is injective between spaces of the same dimension. Hermite derived it, with the error term, from a contour integral generalizing Lagrange’s formula.

Integer matrices and the Hermite normal form

For the general reader. Gaussian elimination solves linear equations over the rational numbers, but it divides, and division is not allowed when the unknowns must be whole numbers. In 1851 Hermite showed that every integer matrix can still be brought to a canonical triangular shape using only invertible integer operations. The result is the integer counterpart of row-reduced echelon form, and it is how computers handle lattices today.

A square integer matrix \(U\) is unimodular if \(\det U=\pm1\); equivalently \(U^{-1}\) also has integer entries. Unimodular matrices form the group \(\mathrm{GL}_n(\mathbb Z)\). Multiplying by \(U\) on the left performs invertible integer row operations: swapping rows, negating a row, and adding an integer multiple of one row to another.

Existence. Work on the first column. While two of its entries are non-zero, subtract an integer multiple of the row with the smaller entry (in absolute value) from the other; this is the Euclidean algorithm, and it ends with a single non-zero entry, equal to \(\pm\gcd\) of the column. Move that row to the top and make the entry positive. Repeat on the lower-right submatrix. Finally, for each column \(j\), reduce the entries above \(h_{jj}\) modulo \(h_{jj}\) by subtracting multiples of row \(j\).

Uniqueness. If \(H\) and \(H'=VH\) are both normal forms with \(V\) unimodular, then \(V=H'H^{-1}\) is upper triangular with positive diagonal entries whose product is \(|\det V|=1\), so all are 1. Suppose \(V\ne I\) and take the lowest row \(i\) with an off-diagonal entry \(v_{ik}\ne0\), \(k>i\) smallest. Then \(h'_{ik}=h_{ik}+v_{ik}h_{kk}\), and both \(h_{ik}\) and \(h'_{ik}\) lie in \([0,h_{kk})\), which forces \(v_{ik}=0\), a contradiction.

The rows of \(A\) and of \(H\) generate the same lattice in \(\mathbb Z^n\), so the theorem gives a canonical basis of every full-rank sublattice of \(\mathbb Z^n\). Two integer matrices generate the same lattice exactly when they have the same Hermite normal form. The diagonal gives the index: \([\mathbb Z^n:L]=\prod_j h_{jj}=|\det A|\). Henry Smith’s normal form of 1861, which also uses column operations, gives the elementary divisors.

The Sorbonne years, 1869–1897

In 1869 Hermite succeeded Jean-Marie Duhamel as professor of analysis at the École Polytechnique and professor at the Faculté des sciences of Paris, the Sorbonne. He gave up the Polytechnique chair in 1876, where Camille Jordan followed him, and kept the Sorbonne chair until 1897. His lectures were celebrated. Henri Poincaré, Émile Picard, Paul Appell and Jacques Hadamard heard him, and his printed Cours d’analyse spread his style. Poincaré described him as the most thorough Platonic realist among the mathematicians he knew, and Hermite wrote to Thomas Stieltjes in 1889 that numbers seemed to him to exist outside us.

His doctoral students included Henri Poincaré, Thomas Stieltjes, Henri Padé, Jules Tannery, Gaston Floquet and Mihailo Petrović. His correspondence was enormous: with Stieltjes, with Gösta Mittag-Leffler (whom he sent to Berlin to study with Weierstrass), with Weierstrass and Kronecker, and with Cayley and Sylvester. Through it he kept French mathematics in close touch with German work in the decades after the war of 1870, when relations between the two countries were bitter.

His later research included linear differential equations with doubly periodic coefficients, notably Lamé’s equation, and continued fractions and approximation. He disliked geometry, and he was famously uneasy with the new set-theoretic analysis; in an 1893 letter to Stieltjes he wrote of turning away in horror from continuous functions without derivatives. He was president of the Académie in 1890. For his seventieth birthday, in December 1892, colleagues and former students held a jubilee in his honour, and he was promoted grand officer of the Légion d’honneur.

Death and legacy

Hermite retired in 1897, and his son-in-law Émile Picard succeeded him at the Sorbonne. (His other daughter married the engineer Georges Forestier.) He died in Paris on 14 January 1901, aged 78, and was buried in the Montparnasse cemetery. His Œuvres, edited by Picard, appeared in four volumes from 1905 to 1917, and his correspondence with Stieltjes was published in 1905.

Legacy

• Number theory: reduction theory and Hermite’s constant began the geometry of numbers, later developed by Minkowski; the finiteness of number fields of bounded discriminant remains a basic theorem.
• Transcendence: his 1873 method, refined by Lindemann, Weierstrass, Hilbert, Siegel and others, is the ancestor of all modern transcendence theory; Padé approximation grew from it.
• Algebra and analysis: Hermitian matrices and the spectral theorem for them; the modular solution of the quintic, which led to Klein’s icosahedron lectures; elliptic and theta functions.
• Applied mathematics and physics: Hermite polynomials in probability (as the orthogonal polynomials of the Gaussian), in quantum mechanics and signal processing; Hermite interpolation and cubic Hermite splines in numerical analysis and computer graphics.
• Teaching: a generation of French analysts, including Poincaré, Picard, Appell and Stieltjes.
• Commemoration: the lunar crater Hermite, near the north pole of the Moon, and the Rue Charles Hermite in Paris.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Charles Hermite, MacTutor History of Mathematics.
2. Charles Hermite, Wikipedia; Charles Hermite, Wikipédia (French).
3. B. Baillaud and H. Bourget (eds.), Correspondance d’Hermite et de Stieltjes, 2 vols., Gauthier-Villars, 1905.
4. É. Picard, Charles Hermite, in Œuvres de Charles Hermite, vol. 1, Gauthier-Villars, 1905.

Mathematics

1. Œuvres de Charles Hermite, ed. É. Picard, 4 vols., Gauthier-Villars, 1905–1917.
2. C. Hermite, Extraits de lettres à M. Jacobi sur différents objets de la théorie des nombres, J. reine angew. Math. 40 (1850).
3. C. Hermite, Sur la résolution de l’équation du cinquième degré, Comptes rendus 46 (1858).
4. C. Hermite, Sur un nouveau développement en série des fonctions, Comptes rendus 58 (1864).
5. C. Hermite, Sur la fonction exponentielle, Comptes rendus 77 (1873).
6. C. Hermite, Sur la formule d’interpolation de Lagrange, J. reine angew. Math. 84 (1878).
7. F. Klein, Vorlesungen über das Ikosaeder, Teubner, 1884.
8. D. Hilbert, Über die Transcendenz der Zahlen e und π, Math. Ann. 43 (1893).
9. I. Niven, A simple proof that π is irrational, Bull. Amer. Math. Soc. 53 (1947).
10. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer, 3rd ed. 1999.
11. H. Cohn and A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, Annals of Math. 170 (2009).
12. T. J. Stieltjes’ letters, “Marvelous cancellations”: T. J. Stieltjes’ letters concerning the zeta function, arXiv:1410.8592.