User:Richard Pinch/sandbox-CZ
ABC conjecture
In mathematics, the ABC conjecture relates the prime factors of two integers to those of their sum. It was proposed by David Masser and Joseph Oesterlé in 1985. It is connected with other problems of number theory: for example, the truth of the ABC conjecture would provide a new proof of Fermat's Last Theorem.
Statement
Define the radical of an integer to be the product of its distinct prime factors
\[ r(n) = \prod_{p|n} p \ . \]
Suppose now that the equation \(A + B + C = 0\) holds for coprime integers \(A,B,C\). The conjecture asserts that for every \(\epsilon > 0\) there exists \(\kappa(\epsilon) > 0\) such that
\[ |A|, |B|, |C| < \kappa(\epsilon) r(ABC)^{1+\epsilon} \ . \]
A weaker form of the conjecture states that
\[ (|A| \cdot |B| \cdot |C|)^{1/3} < \kappa(\epsilon) r(ABC)^{1+\epsilon} \ . \]
If we define
\[ \kappa(\epsilon) = \inf_{A+B+C=0,\ (A,B)=1} \frac{\max\{|A|,|B|,|C|\}}{N^{1+\epsilon}} \ , \]
then it is known that \(\kappa \rightarrow \infty\) as \(\epsilon \rightarrow 0\).
Baker introduced a more refined version of the conjecture in 1998. Assume as before that \(A + B + C = 0\) holds for coprime integers \(A,B,C\). Let \(N\) be the radical of \(ABC\) and \(\omega\) the number of distinct prime factors of \(ABC\). Then there is an absolute constant \(c\) such that
\[ |A|, |B|, |C| < c (\epsilon^{-\omega} N)^{1+\epsilon} \ . \]
This form of the conjecture would give very strong bounds in the method of linear forms in logarithms.
Results
It is known that there is an effectively computable \(\kappa(\epsilon)\) such that
\[ |A|, |B|, |C| < \exp\left({ \kappa(\epsilon) N^{1/3} (\log N)^3 }\right) \ . \]
References
Albert algebra
The set of 3×3 self-adjoint matrices over the octonions with binary operation $$ x \star y = \frac12 (x \cdot y + y \cdot x) \,, $$ where $\cdot$ denotes matrix multiplication.
The operation is commutative but not associative. It is an example of an exceptional Jordan algebra. Because most other exceptional Jordan algebras are constructed using this one, it is often referred to as "the" exceptional Jordan algebra.
References
- A. V. Mikhalev, Gunter F. Pilz, "The Concise Handbook of Algebra", (Springer, 2002) ISBN 0792370724, page 346.
Alternant code
A class of parameterised error-correcting codes which generalise the BCH codes.
An alternant code over $GF(q)$ of length $n$ is defined by a parity check matrix $H$ of alternant form $H_{i,j} = \alpha_j^i y_i$, where the $\alpha_j$ are distinct elements of the extension $GF(q^m)$, the $y_i$ are further non-zero parameters again in the extension $GF(q^m)$ and the indices range as $i$ from 0 to $\delta-1$, $j$ from 1 to $n$.
The parameters of this alternant code are length $n$, dimension $\ge n - m\delta$ and minimum distance $\ge \delta+1$. There exist long alternant codes which meet the Gilbert-Varshamov bound.
The class of alternant codes includes BCH codes, Goppa codes and Srivasta codes.
References
Artin–Schreier polynomial
A polynomial whose roots are used to generate field extensions of prime degree p in characteristic p.
An Artin–Schreier polynomial over a field $F$ is of the form $$ A_\alpha(X) = X^p - X - \alpha $$ for $\alpha \in F$. The function $A : X \mapsto X^p - X$ is $p$-to-one since $A(x) = A(x+1)$. It is in fact $\mathbf{F}_p$-linear on $F$ as a vector space, with kernel the one-dimensional subspace generated by $1_F$, that is, $\mathbf{F}_p$ itself.
Suppose that $F$ is finite of characteristic $F$. The Frobenius map is an automorphism of $p$ and so its inverse, the $p$-th root map is defined everywhere, and $p$-th roots do not generate any non-trivial extensions.
If $F$ is finite, then $A$ is exactly $p$-to-1 and the image of $A$ is a $\mathbf{F}_p$-subspace of codimension 1. There is always some element $\alpha \in F$ not in the image of $A$, and so the corresponding Artin-Schreier polynomial has no root in $F$: it is an irreducible polynomial and the quotient ring $F[X]/\langle A_\alpha(X) \rangle$ is a field which is a degree $p$ extension of $F$. Since finite fields of the same order are unique up to isomorphism, we may say that this is "the" degree $p$ extension of $F$. As before, both roots of the equation lie in the extension, which is thus a splitting field for the equation and hence a Galois extension: in this case the roots are of the form $\beta,~\beta+1, \ldots,\beta+(p-1)$.
Szpiro's conjecture
A conjectural relationship between the conductor and the discriminant of an elliptic curve. In a general form, it is equivalent to the well-known ABC conjecture. It is named for Lucien Szpiro who formulated it in the 1980s.
The conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with minimal discriminant Δ and conductor f, we have
\[ \vert\Delta\vert \leq C(\varepsilon ) \cdot f^{6+\varepsilon }. \, \]
The modified Szpiro conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with invariants c4, c6 and conductor f, we have
\[ \max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon }. \, \]
References
Wiener–Ikehara theorem
A Tauberian theorem relating the behaviour of a real sequence to the analytic properties of the associated Dirichlet series. It is used in the study of arithmetic functions and yields a proof of the Prime number theorem. It was proved by Norbert Wiener and his student Shikao Ikehara in 1932.
Let $F(x)$ be a non-negative, monotonic decreasing function of the positive real variable $x$. Suppose that the Laplace transform
$$
\int_0^\infty F(x)\exp(-xs) dx
$$
converges for $\Re s >1$ to the function $f(s)$ and that $f(s)$ is analytic for $\Re s \ge 1$, except for a simple pole at $s=1$ with residue 1. Then the limit as $x$ goes to infinity of $e^{-x} F(x)$ is equal to 1.
An important number-theoretic application of the theorem is to Dirichlet series of the form $\sum_{n=1}^\infty a(n) n^{-s}$ where $a(n)$ is non-negative. If the series converges to an analytic function in $\Re s \ge b$ with a simple pole of residue $c$ at $s = b$, then $\sum_{n\le X}a(n) \sim c \cdot X^b$.
Applying this to the logarithmic derivative of the Riemann zeta function, where the coefficients in the Dirichlet series are values of the von Mangoldt function, it is possible to deduce the prime number theorem from the fact that the zeta function has no zeroes on the line $\Re (s)=1$.
References
Richard Pinch/sandbox-CZ. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-CZ&oldid=30359