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Todd polynomials

A sequence of polynomials with rational number coefficients associated with Todd classes.

Let $$ H(z; \xi_1,\ldots,\xi_s) = \prod_{i=1}^s \frac{z \xi_i}{1 - \exp(-z\xi_i)} \ . $$ The $m$-th Todd polynomial $T_m(c_1,\ldots,c_m)$ is defined by $T_m(\sigma_1,\ldots,\sigma_m)$ being the coefficient of $z^m$ in the power series expansion of $H(z; \xi_1,\ldots,\xi_m)$ where the $\sigma_i$ are the elementary symmetric functions of the $\xi_i$.

We have $T_1(c_1) = \frac12 c_1$, $T_2(c_1,c_2) = \frac1{12}(c_1^2 + c_2)$, $T_3(c_1,c_2,c_3) = \frac{1}{24}c_1c_2$.

The Todd polynomials are derived from the multiplicative sequence corresponding to the power series $t/(1-e^{-t})$.

References

• Hirzebruch, Friedrich. Topological methods in algebraic geometry, Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel. Reprint of the 2nd, corr. print. of the 3rd ed. [1978] Berlin: Springer-Verlag (1995). ISBN 3-540-58663-6. Zbl 0843.14009.

Sum of divisors function

of a natural number $n$

The sum of the positive integers divisors of a natural number $n$, including $1$ and $n$: $$ \sigma(n) = \sum_{d | n} d \ . $$ More generally, the function $\sigma_k$ is defined as $$ \sigma_k(n) = \sum_{d | n} d^k \ . $$ so that $\sigma = \sigma_1$ and the number of divisors function $\tau = \sigma_0$.

These are multiplicative arithmetic functions with Dirichlet series $$ \sum_{n=1}^\infty \sigma_k(n) n^{-s} = \prod_p \left({(1-p^{-s})(1-p^{k-s}) }\right)^{-1} = \zeta(s) \zeta(s-k)\ . $$

The average order of $\sigma(n)$ is given by $$ \sum_{n \le x} \sigma(n) = \frac{\pi^2}{12} x^2 + O(x \log x) \ . $$

There are a number of well-known classes of number characterised by their divisor sums.

A perfect number $n$ is the sum of its aliquot divisors (those divisors other than $n$ itself), so $\sigma(n) = 2n$. The even perfect numbers are characterised in terms of Mersenne primes $P = 2^p-1$ as $n = 2^{p-1}.P$: it is not known if there are any odd perfect numbers. An almost perfect number $n$ similarly has the property that $\sigma(n) = 2n-1$: these include the powers of 2. A quasiperfect number is defined by $\sigma(n) = 2n+1$: it is known known if any exists. See also Descartes number.

References

• Kishore, Masao. "On odd perfect, quasiperfect, and odd almost perfect numbers". Mathematics of Computation 36 (1981) 583–586. ISSN 0025-5718. Zbl 0472.10007
• G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics 46, Cambridge University Press (1995) ISBN 0-521-41261-7 Zbl 0831.11001

Exponential law

A term with a variety of meanings.

A process which changes by a given factor over a fixed time period.

A property of the exponential function, such as $\exp(x+y) = \exp(x)\exp(y)$ or of a power, such as $x^{yz} = (x^y)^z$.

A property of sets of maps between sets, such as $\mathrm{Map}(X\times Y,Z) \leftrightarrow \mathrm{Map}(X, \mathrm{Map}(Y,Z))$, see Exponential law for sets.

A property of sets of continuous maps between topological spaces, see Exponential law (in topology).

Exponential law for sets

The correspondence between the sets $A^{B \times C}$ and $(A^B)^C$, where $X^Y$ denotes the set of all maps from the set $Y$ to the set $X$. Given $f \in A^{B \times C}$, that is $F : B \times C \rightarrow A$, and given $c \in C$, let $f_c$ denote the map $f_c : B \rightarrow A$ by $f_c : b \mapsto f(b,c)$. Then $c \mapsto f_c$ defines a map from $A^{B \times C} \rightarrow (A^B)^C$. In the opposite direction, let $G \in (A^B)^C$. Given $b \in B$ and $c \in C$, define $g(b,c)$ to be $G(c)$ applied to $b$. Then $G \mapsto g$ defines a map from $(A^B)^C \rightarrow A^{B \times C}$. These two correspondences are mutually inverse.

In computer science, this operation is known as "Currying" after Haskell Curry (1900-1982).

In category-theoretic terms, the exponential law makes the category of sets a Cartesian-closed category.

In cardinal arithmetic, this corresponds to the formula $$ \mathfrak{a}^{\mathfrak{b}\mathfrak{c}} = (\mathfrak{a}^{\mathfrak{b}})^{\mathfrak{c}} \ . $$

There are further correspondences relating the sets $A^{B \coprod C}$ and $A^B \times A^C$ and the sets $(A \times B)^C$ and $A^C \times B^C$, corresponding to the formulae in cardinal arithmetic $\mathfrak{a}^{\mathfrak{b} + \mathfrak{c}} = \mathfrak{a}^{\mathfrak{b}}\mathfrak{a}^{\mathfrak{c}}$ and $(\mathfrak{a} \mathfrak{b})^{\mathfrak{c}} = \mathfrak{a}^{\mathfrak{c}} \mathfrak{a}^{\mathfrak{c}}$.

References

• Benjamin C. Pierce, Basic Category Theory for Computer Scientists, MIT Press (1991) ISBN 0262660717
• Paul Taylor, Practical Foundations of Mathematics, Cambridge Studies in Advanced Mathematics 59, Cambridge University Press (1999) ISBN 0-521-63107-6

Matrix multiplication

A binary operation on compatible matrices over a ring $R$. There are several such operations.

Cayley multiplication

Most usually what is referred to as "matrix multiplication". The product of an $m \times n$ matrix $A$ and an $n \times p$ matrix $B$ is the $m \times p$ matrix $AB$ with entries $$ (AB)_{ik} = \sum_{j=1}^n a_{ij} b_{jk}\ ,\ \ i=1,\ldots,m\,\ j=1,\ldots,p\,. $$

The multiplication corresponds to composition of linear maps. If $A$ is the matrix of a linear map $\alpha : R^m \rightarrow R^n$ and $B$ is the matrix of a linear map $\beta : R^n \rightarrow R^p$, then $AB$ is the matrix of the linear map $\alpha\beta : R^m \rightarrow R^p$.

Hadamard multiplication

The Hadamard product, or Schur product, of two $m \times n$ matrices $A$ and $B$ is the $m \times n$ matrix $AB$ with $$ (A \circ B)_{ij} = a_{ij} b_{ij}\ ,\ \ i=1,\ldots,m\,\ j=1,\ldots,n\,. $$

Kronecker multiplication

The Kronecker product, also tensor product or direct product, of an $m \times n$ matrix $A$ and an $p \times q$ matrix $B$ is the $mp \times nq$ matrix $AB$ with entries $$ (A \otimes B)_{(i-1)p+k,(j-1)q+l} = a_{ij} b_{kl},\ \ i=1,\ldots,m\,\ j=1,\ldots,n\,\ k=1,\ldots,p\,\ l=1,\ldots,q\,. $$

References

• Gene H. Golub, Charles F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences 3, JHU Press (2013) ISBN 1421407949
• James E. Gentle, Matrix Algebra: Theory, Computations, and Applications in Statistics, Springer Texts in Statistics, Springer (2007) ISBN 0-387-70872-3
• Manfred Schroeder, Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, Springer (2008) ISBN 3-540-85297-2

Multiplicative sequence

Also m-sequence, a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology.

Definition

Let $K_n$ be polynomials over a ring $A$ in indeterminates $p_1,\ldots$ weighted so that $p_i$ has weight $i$ (with $p_0=1$) and all the terms in $K_n$ have weight $n$ (so that $K_n$ is a polynomial in $p_1,\ldots,p_n$). The sequence $K_n$ is multiplicative if an identity

$$\sum_i p_i z^i = \sum p'_i z^i \cdot \sum_i p''_i z^i $$

implies

$$\sum_i K_i(p_1,\ldots,p_i) z^i = \sum_j K_j(p'_1,\ldots,p'_j) z^j \cdot \sum_k K_k(p''_1,\ldots,p''_k) z^k . $$ The power series

$$\sum K_n(1,0,\ldots,0) z^n $$

is the characteristic power series of the $K_n$. A multiplicative sequence is determined by its characteristic power series $Q(z)$, and every power series with constant term 1 gives rise to a multiplicative sequence.

To recover a multiplicative sequence from a characteristic power series $Q(z)$ we consider the coefficient of z^j in the product

$$ \prod_{i=1}^m Q(\beta_i z) $$

for any $m>j$. This is symmetric in the $\beta_i$ and homogeneous of weight j: so can be expressed as a polynomial $K_j(p_1,\ldots,p_j)$ in the elementary symmetric functions $p$ of the $\beta$. Then $K_j$ defines a multiplicative sequence.

Examples

As an example, the sequence $K_n = p_n$ is multiplicative and has characteristic power series $1+z$.

Consider the power series

$$ Q(z) = \frac{\sqrt z}{\tanh \sqrt z} = 1 - \sum_{k=1}^\infty (-1)^k \frac{2^{2k}}{(2k)!} B_k z^k $$ where $B_k$ is the $k$-th Bernoulli number. The multiplicative sequence with $Q$ as characteristic power series is denoted $L_j(p_1,\ldots,p_j)$.

The multiplicative sequence with characteristic power series

$$ Q(z) = \frac{2\sqrt z}{\sinh 2\sqrt z} $$

is denoted $A_j(p_1,\ldots,p_j)$.

The multiplicative sequence with characteristic power series

$$Q(z) = \frac{z}{1-\exp(-z)} = 1 + \frac{x}{2} - \sum_{k=1}^\infty (-1)^k \frac{B_k}{(2k)!} z^{2k} $$ is denoted $T_j(p_1,\ldots,p_j)$: the Todd polynomials.

Genus

The genus of a multiplicative sequence is a ring homomorphism, from the cobordism ring of smooth oriented compact manifolds to another ring, usually the ring of rational numbers.

For example, the Todd genus is associated to the Todd polynomials $T_j$ with characteristic power series $$\frac{z}{1-\exp(-z)}$$ and the L-genus is associated to the polynomials $L_j$ with charac\teristic polynomial $$\frac{\sqrt z}{\tanh \sqrt z} . $$

References

• Hirzebruch, Friedrich. Topological methods in algebraic geometry, Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel. Reprint of the 2nd, corr. print. of the 3rd ed. [1978] (Berlin: Springer-Verlag, 1995). ISBN 3-540-58663-6. Zbl 0843.14009.

Nagao's theorem

A result, named after Hirosi Nagao, about the structure of the group of 2-by-2 invertible matrices over the ring of polynomials over a field. It has been extended by Serre to give a description of the structure of the corresponding matrix group over the coordinate ring of a projective curve.

Nagao's theorem

For a general ring $R$ we let $GL_2(R)$ denote the group of invertible 2-by-2 matrices with entries in $R$, and let $R^*$ denote the group of units of $R$, and let

$$ B(R) = \left\lbrace{ \left({\begin{array}{*{20}c} a & b \\ 0 & d \end{array}}\right) : a,d \in R^*, ~ b \in R }\right\rbrace \ . $$

Then $B(R)$ is a subgroup of $GL_2(R)$.

Nagao's theorem states that in the case that $R$ is the ring $K[t]$ of polynomials in one variable over a field $K$, the group $GL_2(R)$ is the amalgamated product of $GL_2(K)$ and $B(K[t])$ over their intersection $B(K)$.

Serre's extension

In this setting, $C$ is a smooth projective curve over a field $K$. For a closed point $P$ of $C$ let $R$ be the corresponding coordinate ring of $C$ with $P$ removed. There exists a graph of groups $(G,T)$ where $T$ is a tree with at most one non-terminal vertex, such that $GL_2(R)$ is isomorphic to the fundamental group $\pi_1(G,T)$.

References

• Mason, A.. "Serre's generalization of Nagao's theorem: an elementary approach". Transactions of the American Mathematical Society 353 (2001) 749–767. DOI 10.1090/S0002-9947-00-02707-0 Zbl 0964.20027.
• Nagao, Hirosi. "On $GL(2, K[x])$". J. Inst. Polytechn., Osaka City Univ., Ser. A 10 (1959) 117–121. MR0114866. Zbl 0092.02504.
• Serre, Jean-Pierre. Trees. (Springer, 2003) ISBN 3-540-44237-5.

Erdős–Wintner theorem

A result in probabilistic number theory characterising those additive functions that possess a limiting distribution.

Limiting distribution

A distribution function $F$ is a non-decreasing function from the real numbers to the unit interval [0,1] which is right-continuous and has limits 0 at $-\infty$ and 1 at $+\infty$.

Let $f$ be a complex-valued function on natural numbers. We say that $F$ is a limiting distribution for $f$ if $F$ is a distribution function and the sequence $F_N$ defined by

$$ F_n(t) = \frac{1}{N} | \{n \le N : |f(n)| \le t \} | $$

converges weakly to $F$.

Statement of the theorem

Let $f$ be an additive function. There is a limiting distribution for $f$ if and only if the following three series converge: $$ \sum_{|f(p)|>1} \frac{1}{p} \,,\ \sum_{|f(p)|\le1} \frac{f(p)}{p} \,,\ \sum_{|f(p)|\le1} \frac{f(p)^2}{p} \ . $$

When these conditions are satisfied, the distribution is given by $$ F(t) = \prod_p \left({1 - \frac{1}{p} }\right) \cdot \left({1 + \sum_{k=1}^\infty p^{-k}\exp(i t f(p)^k) }\right) \ . $$

References

• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. Handbook of number theory I. Dordrecht: Springer-Verlag (2006). pp. 564–566. ISBN 1-4020-4215-9. Zbl 1151.11300
• Tenenbaum, Gérald Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics 46. Cambridge University Press (1995). ISBN 0-521-41261-7. Zbl 0831.11001

Brauer–Wall group

A group classifying graded central simple algebras over a field. It was first defined by Wall (1964) as a generalisation of the Brauer group.

The Brauer group $\mathrm{B}(F)$ of a field $F$ is defined on the isomorphism classes of central simple algebras over F. The analogous construction for $\mathbf{Z}/2$-graded algebras defines the Brauer–Wall group $\mathrm{BW}(F)$.[Lam (2005) pp.98–99]

Properties

• The Brauer group $\mathrm{B}(F)$ injects into $\mathrm{BW}(F)$ by mapping a CSA $A$ to the graded algebra which is $A$ in grade zero.

There is an exact sequence $$ 0 \rightarrow \mathrm{B}(F) \rightarrow \mathrm{BW}(F) \rightarrow Q(F) \rightarrow 0 $$ where $Q(F)$ is the group of graded quadratic extensions of $F$, defined as $\mathbf{Z}/2 \times F^*/(F^*)^2$ with multiplication $(e,x)(f,y) = (e+f,(-1)^{ef}xy$. The map from W to BW is the Clifford invariant defined by mapping an algebra to the pair consisting of its grade and determinant.

There is a map from the additive group of the Witt–Grothendieck ring to the Brauer–Wall group obtained by sending a quadratic space to its Clifford algebra. The map factors through the Witt group[Lam (2005) p.113] which has kernel $I^3$, where $I$ is the fundamental ideal of $W(F)$.[Lam (2005) p.115]

Examples

• $\mathrm{BW}(\mathbf{R})$ is isomorphic to $\mathbf{Z}/8$. This is an algebraic aspect of Bott periodicity.

References

• Lam, Tsit-Yuen, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, (American Mathematical Society, 2005) ISBN 0-8218-1095-2 MR2104929, Zbl 1068.11023
• Wall, C. T. C., "Graded Brauer groups", Journal für die reine und angewandte Mathematik 213 (1964) 187–199, ISSN 0075-4102, Zbl 0125.01904, MR0167498

Factor system

A function on a group giving the data required to construct an algebra. A factor system constitutes a realisation of the cocycles in the second cohomology group in group cohomology.

Let $G$ be a group and $L$ a field on which $G$ acts as automorphisms. A cocycle or factor system is a map $c : G \times G \rightarrow L^*$ satisfying $$ c(h,k)^g c(hk,g) = c(h,kg) c(k,g) \ . $$

Cocycles are equivalent if there exists some system of elements $a : G \rightarrow L^*$ with $$ c'(g,h) = c(g,h) (a_g^h a_h a_{gh}^{-1}) \ . $$

Cocycles of the form $$ c(g,h) = a_g^h a_h a_{gh}^{-1} $$ are called split. Cocycles under multiplication modulo split cocycles form a group, the second cohomology group $H^2(G,L^*)$.

Crossed product algebras

Let us take the case that $G$ is the Galois group of a field extension $L/K$. A factor system $c$ in $H^2(G,L^*)$ gives rise to a crossed product algebra $A$, which is a $K$-algebra containing $L$ as a subfield, generated by the elements $\lambda \in L$ and $u_g$ with multiplication $$ \lambda u_g = u_g \lambda^g \ , $$ $$ u_g u_h = u_{gh} c(g,h) \ . $$ Equivalent factor systems correspond to a change of basis in $A$ over $K$. We may write $$ A = (L,G,c) \ .$$

Every central simple algebra over$K$ that splits over $L$ arises in this way. The tensor product of algebras corresponds to multiplication of the corresponding elements in$H^2$. We thus obtain an identification of the Brauer group, where the elements are classes of CSAs over $K$, with $H^2$.[Saltman (1999) p.44]

Cyclic algebra

Let us further restrict to the case that $L/K$ is cyclic with Galois group $G$ of order $n$ generated by $t$. Let $A$ be a crossed product $(L,G,c)$ with factor set $c$. Let $u=u_t$ be the generator in $A$ corresponding to $t$. We can define the other generators $$ u_{t^i} = u^i $$ and then we have $u^n = a$ in $K$. This element $a$ specifies a cocycle $c$ by $$ c(t^i,t^j) = \begin{cases} 1 & \text{if } i+j < n, \\ a & \text{if } i+j \ge n. \end{cases} $$

It thus makes sense to denote $A$ simply by $(L,t,a)$. However $a$ is not uniquely specified by $A$ since we can multiply $u$ by any element $\lambda$ of $L^*$ and then $a$ is multiplied by the product of the conjugates of λ. Hence $A$ corresponds to an element of the norm residue group $(K^*/N_{L/K}L^*$. We obtain the isomorphisms $$ \mathop{Br}(L/K) \equiv K^*/\mathrm{N}_{L/K} L^* \equiv \mathrm{H}^2(G,L^*) \ . $$

References

• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Universitext. Translated from the German by Silvio Levy. With the collaboration of the translator. Springer-Verlag. ISBN 978-0-387-72487-4. Zbl 1130.12001.
• Saltman, David J. (1999). Lectures on division algebras. Regional Conference Series in Mathematics 94. Providence, RI: American Mathematical Society. ISBN 0-8218-0979-2. Zbl 0934.16013.