On the rational approximation to linear combinations of powers

We have seen in the last chapter that the field of complex numbers admits no further algebraic extensions. Within the field of complex numbers, however, there are many numbers that are not algebraic over $$\mathbb {Q}$$ . In fact, the algebraically closed field of all algebraic numbers in $$\mathbb {C}$$ is a countable set, for you can check that the algebraic numbers over $$\mathbb {Q}$$ that have a minimal polynomial of degree n are countable. Letting n vary gives a countable collection of countable sets, which is therefore countable.

where the Li(x) are homogeneous polynomials of degree 1 with coefficients which are algebraic numbers. One of the most interesting cases arises when the following two conditions are satisfied. (1) The Li(x) never vanish simultaneously at any of V. (2) Under the mapping a of V again becomes a of V. Generally speaking the only points which interest us are those which can be expressed with coordinates which are algebraic numbers. = (to, I, * * * n is such a point, the number field generated by the ratios ti/tj will be the smallest field in which is rational. The degree of this field over the rational numbers, we shall call the degree of rationality of P. Suppose now that conditions (1) and (2) are satisfied, then by iterating the mapping, a given of V will generate an infinite sequence = Po, P1, P2, *** of points on the variety. This sequence will consist either of distinct points or else there will be repetitions. In the latter case the sequence ultimately becomes periodic, and we shall describe this situation by saying P is an exceptional point. One of the results established can now be stated, namely, If we exclude the case in which the mapping is linear, then there will be at most a finite number of exceptional points with a given degree of rationality. We can illustrate this result by means of the well known Weierstrass function P(Z1 g2, g3), where g2 and g3 are algebraic numbers. A complex number 'a' is called a point if for some integer n, which is not zero, na is a period. The values of p(z) at the division points are called values, and it is easy to see that these are algebraic numbers. By comparing the elliptic function with the exponential function, we see that these division values are in some respects analogous to roots of unity, and it is therefore natural to inquire how they are distributed among the number fields. A special case of the general theorem stated above, shows that if we limit ourselves to number fields whose absolute degrees are bounded by some given integer, then amongst all these fields there will be only a finite number of division values. These facts are consequences of Theorem 2 which is an inequality proved under very general conditions. This inequality has another application (not discussed here), namely 167

A fundamental problem of experimental algebraic number theory is that of determining the unitand class group of an algebraic number field K. To solve this problem for large classes of number fields, effective algorithms are needed. Such algorithms mostly rest on methods from geometry of numbers, as this is the case with the method of Pohst and Zassenhaus. They require knowledge of an integral basis for K/~. Such bases of number fields canbe constructed in an efficient manner by applying an algorithm of Ford and Zassenhaus° Here local considerations play an important role. That is why one has to perform calculations not only over the completion of Q with respect to ordinary absolute value, i.e. over the real field IR, but at the same time also over the completions of Q with respect to the p-adic valuations, i.e. over the p-adic fields Up. Furthermore, a p-adic analogue of the complex number field ~ the latter being the ~]g~h~ closure of ~rR, is the completion Cp of the algebraic closure ~p of ~p. In this connection we recall that Hasse, in his fundamental book on algebraic number theory, uses a simultaneous treatment of the completions IR and @p as a foundation of the arithmetic in algebraic number fields.

Chapter 54 - David Hilbert, report on algebraic number fields (‘Zahlbericht’) (1897)

It is proved that the field of complex algebraic numbers has an isomorphic presentation computable in polynomial time. A similar fact is proved for the ordered field of real algebraic numbers. The constructed polynomially computable presentations are based on a natural presentation of algebraic numbers by rational polynomials. Also new algorithms for computing values of polynomials on algebraic numbers and for solving equations in one variable with algebraic coefficients are presented.

We introduce a canonical representation of call options, and propose a solution to two open problems in option pricing theory. The first problem was posed by (Kassouf, 1969, pg. 694) seeking “theoretical substantiation” for his robust option pricing power law which eschewed assumptions about risk attitudes, rejected risk neutrality, and made no assumptions about stock price distribution. The second problem was posed by (Scott, 1987, pp. 423-424) who could not find a unique solution to the call option price in his option pricing model with stochastic volatility–without appealing to an equilibrium asset pricing model by Hull and White (1987), and concluded: “[w]e cannot determine the price of a call option without knowing the price of another call on the same stock”. First, we show that under certain conditions derivative assets are superstructures of the underlying. Hence any option pricing or derivative pricing model in a given number field, based on an anticipating variable in an extended field, with coefficients in a subfield containing the underlying, is admissible for market timing. For the anticipating variable is an algebraic number that generates the subfield in which it is the root of an equation. Accordingly, any polynomial which satisfies those criteria is admissible for price discovery and or market timing. Therefore, at least for empirical purposes, elaborate models of mathematical physics or otherwise are unnecessary for pricing derivatives because much simpler adaptive polynomials in suitable algebraic numbers are functionally equivalent. Second, we prove, analytically, that Kassouf (1969) power law specification for option pricing is functionally equivalent to Black and Scholes (1973); Merton (1973) in an algebraic number field containing the underlying. In fact, we introduce a canonical polynomial representation theory of call option pricing convex in time to maturity, and algebraic number of the underlying–with coefficients based on observables in a subfield. Thus, paving the way for Wold decomposition of option prices, and subsequently laying a theoretical foundation for a GARCH option pricing model. Third, our canonical representation theory has an inherent regenerative multifactor decomposition of call option price that (1) induces a duality theorem for call option prices, and (2) permits estimation of risk factor exposure for Greeks by standard [polynomial] regression procedures. Thereby providing a theoretical (a) basis for option pricing of Greeks, and (b) solving Scott’s dual call option problem a fortiori with our duality theory in tandem with Riesz representation theory. Fourth, when the Wold decomposition procedure is applied we are able to construct an empirical pricing kernel for call option based on residuals from a model of risk exposure to persistent and transient risk factors. <div><br><div>Keywords: number theory; price discovery; derivatives pricing; asset pricing; canonical representation; Wold decomposition; empirical pricing kernel; option Greeks; dual option pricing</div></div>

We introduce a canonical representation of call options, and propose a solution to two open problems in option pricing theory. The first problem was posed by (Kassouf, 1969, pg. 694) seeking “theoretical substantiation” for his robust option pricing power law which eschewed assumptions about risk attitudes, rejected risk neutrality, and made no assumptions about stock price distribution. The second problem was posed by (Scott, 1987, pp. 423-424) who could not find a unique solution to the call option price in his option pricing model with stochastic volatility–without appealing to an equilibrium asset pricing model by Hull and White (1987), and concluded: “[w]e cannot determine the price of a call option without knowing the price of another call on the same stock”. First, we show that under certain conditions derivative assets are superstructures of the underlying. Hence any option pricing or derivative pricing model in a given number field, based on an anticipating variable in an extended field, with coefficients in a subfield containing the underlying, is admissible for market timing. For the anticipating variable is an algebraic number that generates the subfield in which it is the root of an equation. Accordingly, any polynomial which satisfies those criteria is admissible for price discovery and or market timing. Therefore, at least for empirical purposes, elaborate models of mathematical physics or otherwise are unnecessary for pricing derivatives because much simpler adaptive polynomials in suitable algebraic numbers are functionally equivalent. Second, we prove, analytically, that Kassouf (1969) power law specification for option pricing is functionally equivalent to Black and Scholes (1973); Merton (1973) in an algebraic number field containing the underlying. In fact, we introduce a canonical polynomial representation theory of call option pricing convex in time to maturity, and algebraic number of the underlying–with coefficients based on observables in a subfield. Thus, paving the way for Wold decomposition of option prices, and subsequently laying a theoretical foundation for a GARCH option pricing model. Third, our canonical representation theory has an inherent regenerative multifactor decomposition of call option price that (1) induces a duality theorem for call option prices, and (2) permits estimation of risk factor exposure for Greeks by standard [polynomial] regression procedures. Thereby providing a theoretical (a) basis for option pricing of Greeks, and (b) solving Scott’s dual call option problem a fortiori with our duality theory in tandem with Riesz representation theory. Fourth, when the Wold decomposition procedure is applied we are able to construct an empirical pricing kernel for call option based on residuals from a model of risk exposure to persistent and transient risk factors. <div><br><div>Keywords: number theory; price discovery; derivatives pricing; asset pricing; canonical representation; Wold decomposition; empirical pricing kernel; option Greeks; dual option pricing</div></div>

In this paper, we obtain an explicit form of the currently best known inequality for linear forms in the logarithms of algebraic numbers. The results complete our previous investigations (Bull. Austral. Math. Soc. 15 (1976), 33–57) which were conditional on a certain independence condition on the algebraic numbers. The extra work needed to obtain unconditional results centres on the properties of multiplicative relations in number fields. In particular, we show that a set of multiplicatively dependent algebraic numbers always satisfies a relation with relatively small exponents.

This chapter establishes some essential foundational results in the subject of algebraic number theory beyond what was already in Basic Algebra.Section 1 puts matters in perspective by examining what was proved in Chapter I for quadratic number fields and picking out questions that need to be addressed before one can hope to develop a comparable theory for number fields of degree greater than 2.Sections 2–4 concern the field discriminant of a number field. Section 2 contains the definition of discriminant, as well as some formulas and examples. The main result of Section 3 is the Dedekind Discriminant Theorem. This concerns how prime ideals (p) in Z split when extended to the ideal (p)R in the ring of integers R of a number field. The theorem says that ramification, i.e, the occurrence of some prime ideal factor in R to a power greater than 1, occurs if and only if p divides the field discriminant. The theorem is proved only in a very useful special case, the general case being deferred to Chapter VI. The useful special case is obtained as a consequence of Kummer's criterion, which relates the factorization modulo p of irreducible monic polynomials in Z[X] to the question of the splitting of the ideal (p)R. Section 4 gives a number of examples of the theory for number fields of degree 3.Section 5 establishes the Dirichlet Unit Theorem, which describes the group of units in the ring of algebraic integers in a number field. The torsion subgroup is the subgroup of roots of unity, and it is finite. The quotient of the group of units by the torsion subgroup is a free abelian group of a certain finite rank. The proof is an application of the Minkowski Lattice-Point Theorem.Section 6 concerns class numbers of algebraic number fields. Two nonzero ideals I and J in the ring of algebraic integers of a number field are equivalent if there are nonzero principal ideals (a) and (b) with (a)I = (b)J . It is relatively easy to prove that the set of equivalence classes has a group structure and that the order of this group, which is called the class number, is finite. The class number is 1 if and only if the ring is a principal ideal domain. One wants to be able to compute class numbers, and this easy proof of finiteness of class numbers is not helpful toward this end. Instead, one applies theMinkowski Lattice-Point Theorem a second time, obtaining a second proof of finiteness, one that has a sharp estimate for a finite set of ideals that need to be tested for equivalence. Some examples are provided. A by-product of the sharp estimate is Minkowski's theorem that the field discriminant of any number field other than Q is greater than 1. In combination with the Dedekind Discriminant Theorem, this result shows that there always exist ramified primes over Q.

This chapter establishes some essential foundational results in the subject of algebraic number theory beyond what was already in Basic Algebra. Section 1 puts matters in perspective by examining what was proved in Chapter I for quadratic number fields and picking out questions that need to be addressed before one can hope to develop a comparable theory for number fields of degree greater than 2. Sections 2–4 concern the field discriminant of a number field. Section 2 contains the definition of discriminant, as well as some formulas and examples. The main result of Section 3 is the Dedekind Discriminant Theorem. This concerns how prime ideals $(p)$ in $\mathbb{Z}$ split when extended to the ideal $(p)R$ in the ring of integers $R$ of a number field. The theorem says that ramification, i.e, the occurrence of some prime ideal factor in $R$ to a power greater than 1, occurs if and only if $p$ divides the field discriminant. The theorem is proved only in a very useful special case, the general case being deferred to Chapter VI. The useful special case is obtained as a consequence of Kummer's criterion, which relates the factorization modulo $p$ of irreducible monic polynomials in $\mathbb{Z}[X]$ to the question of the splitting of the ideal $(p)R$. Section 4 gives a number of examples of the theory for number fields of degree 3. Section 5 establishes the Dirichlet Unit Theorem, which describes the group of units in the ring of algebraic integers in a number field. The torsion subgroup is the subgroup of roots of unity, and it is finite. The quotient of the group of units by the torsion subgroup is a free abelian group of a certain finite rank. The proof is an application of the Minkowski Lattice-Point Theorem. Section 6 concerns class numbers of algebraic number fields. Two nonzero ideals $I$ and $J$ in the ring of algebraic integers of a number field are equivalent if there are nonzero principal ideals $(a)$ and $(b)$ with $(a)I=(b)J$. It is relatively easy to prove that the set of equivalence classes has a group structure and that the order of this group, which is called the class number, is finite. The class number is 1 if and only if the ring is a principal ideal domain. One wants to be able to compute class numbers, and this easy proof of finiteness of class numbers is not helpful toward this end. Instead, one applies the Minkowski Lattice-Point Theorem a second time, obtaining a second proof of finiteness, one that has a sharp estimate for a finite set of ideals that need to be tested for equivalence. Some examples are provided. A by-product of the sharp estimate is Minkowski's theorem that the field discriminant of any number field other than $\mathbb{Q}$ is greater than 1. In combination with the Dedekind Discriminant Theorem, this result shows that there always exist ramified primes over $\mathbb{Q}$.

The paper deals with several criteria for the transcendence of infinite products of the form $$\prod\limits_{n = 1}^\infty {[{b_n}{a^{{a_n}}}]/{b_n}{a^{{a_n}}}} $$ where α > 1 is a positive algebraic number having a conjugate α* such that α ≠ |α*| > 1, {a n } =1 ∞ and {b n } =1 ∞ are two sequences of positive integers with some specific conditions. The proofs are based on the recent theorem of Corvaja and Zannier which relies on the Subspace Theorem (P.Corvaja, U.Zannier: On the rational approximation to the powers of an algebraic number: solution of two problems of Mahler and Mend`es France, Acta Math. 193, (2004), 175–191).

A complex number α is said to be an algebraic number if there is a non-zero polynomial \(f(x) \in \mathbb{Q}[x]\) such that f(α) = 0. Given an algebraic number α, there exists a unique irreducible monic polynomial \(P(x) \in \mathbb{Q}[x]\) such that P(α) = 0. This is called the minimal polynomial of α. The set of all algebraic numbers denoted by \(\overline{\mathbb{Q}}\) is a subfield of the field of complex numbers. A complex number which is not algebraic is said to be transcendental.

We prove that the field of complex algebraic numbers and the ordered field of real algebraic numbers have isomorphic presentations computable in polynomial time. For these presentations, new algorithms are found for evaluation of polynomials and solving equations of one unknown. It is proved that all best known presentations for these fields produce polynomially computable structures or quotient-structures such that there exists an isomorphism between them polynomially computable in both directions.

Let Q denote the rational number field and α be an algebraic number of degree s. Then the algebraic number field F s = Q(α) is the field given by the polynomials in α of degree < s with rational coefficients.

Comments on the height reducing property II