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Carlitz–Wan Conjecture
In mathematics, the Carlitz–Wan conjecture classifies the possible degrees of exceptional polynomials over a finite field ''F''''q'' of ''q'' elements. A polynomial ''f''(''x'') in ''F''''q'' 'x''of degree ''d'' is called exceptional over ''F''''q'' if every irreducible factor (differing from ''x'' − ''y'') or (''f''(''x'') − ''f''(''y''))/(''x'' − ''y'')) over ''F''''q'' becomes reducible over the algebraic closure of ''F''''q''. If ''q'' > ''d''4, then ''f''(''x'') is exceptional if and only if ''f''(''x'') is a permutation polynomial over ''F''''q''. The Carlitz–Wan conjecture states that there are no exceptional polynomials of degree ''d'' over ''F''''q'' if gcd(''d'', ''q'' − 1) > 1. In the special case that ''q'' is odd and ''d'' is even, this conjecture was proposed by Leonard Carlitz (1966) and proved by Fried, Guralnick, and Saxl (1993). The general form of the Carlitz–Wan conjecture was proposed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see order of a polynomial (other)). For example, the polynomial 7x^2y^3 + 4x - 9, which can also be written as 7x^2y^3 + 4x^1y^0 - 9x^0y^0, has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (x+1)^2 - (x-1)^2, one can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky (1972) pp.74-76 or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field ''K'' is unique up to an isomorphism that fixes every member of ''K''. Because of this essential uniqueness, we often speak of ''the'' algebraic closure of ''K'', rather than ''an'' algebraic closure of ''K''. The algebraic closure of a field ''K'' can be thought of as the largest algebraic extension of ''K''. To see this, note that if ''L'' is any algebraic extension of ''K'', then the algebraic closure of ''L'' is also an algebraic closure of ''K'', and so ''L'' is contained within the algebraic closure of ''K''. The algebraic closure of ''K'' is also the smallest algebraically closed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Polynomial
In mathematics, a permutation polynomial (for a given ring) is a polynomial that acts as a permutation of the elements of the ring, i.e. the map x \mapsto g(x) is a bijection. In case the ring is a finite field, the Dickson polynomials, which are closely related to the Chebyshev polynomials, provide examples. Over a finite field, every function, so in particular every permutation of the elements of that field, can be written as a polynomial function. In the case of finite rings Z/''n''Z, such polynomials have also been studied and applied in the interleaver component of error detection and correction algorithms. Single variable permutation polynomials over finite fields Let be the finite field of characteristic , that is, the field having elements where for some prime . A polynomial with coefficients in (symbolically written as ) is a ''permutation polynomial'' of if the function from to itself defined by c \mapsto f(c) is a permutation of . Due to the finiteness of , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (mathematics)
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonard Carlitz
Leonard Carlitz (December 26, 1907 – September 17, 1999) was an American mathematician. Carlitz supervised 44 doctorates at Duke University and published over 770 papers. Chronology * 1907 Born Philadelphia, PA, USA * 1927 BA, University of Pennsylvania * 1930 PhD, University of Pennsylvania, 1930 under Howard Mitchell, who had studied under Oswald Veblen at Princeton * 1930–31 at Caltech with E. T. Bell * 1931 married Clara Skaler * 1931–32 at Cambridge with G. H. Hardy * 1932 Joined the faculty of Duke University where he served for 45 years * 1938 to 1973 Editorial Board Duke Mathematical Journal (Managing Editor from 1945.) * 1939 Birth of son Michael * 1940 Supervision of his first doctoral student E. F. Canaday, awarded 1940 * 1945 Birth of son Robert * 1964 First James B. Duke Professor in Mathematics * 1977 Supervised his 44th and last doctoral student, Jo Ann Lutz, awarded 1977 * 1977 Retired * 1990 Death of wife Clara, after 59 years of marriage * 1999 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... was 0.754. External links * Mathematics journals Publications established in 1963 English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daqing Wan
Daqing Wan (born 1964 in China) is a Chinese mathematician working in the United States. He received his Ph.D. from the University of Washington in Seattle in 1991, under the direction of Neal Koblitz. Since 1997, he has been on the faculty of mathematics at the University of California at Irvine; he has also held visiting positions at the Institute for Advanced Study in Princeton, New Jersey, Pennsylvania State University, the University of Rennes, the Mathematical Sciences Research Institute in Berkeley, California, and the Chinese Academy of Sciences in Beijing. His primary interests include number theory and arithmetic algebraic geometry, particularly zeta functions over finite fields. He is known for his proof of Dwork's conjecture that the p-adic unit root zeta function attached to a family of varieties over a finite field of characteristic p is p-adic meromorphic In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hendrik Lenstra
Hendrik Willem Lenstra Jr. (born 16 April 1949, Zaandam) is a Dutch mathematician. Biography Lenstra received his doctorate from the University of Amsterdam in 1977 and became a professor there in 1978. In 1987 he was appointed to the faculty of the University of California, Berkeley; starting in 1998, he divided his time between Berkeley and the University of Leiden, until 2003, when he retired from Berkeley to take a full-time position at Leiden. Three of his brothers, Arjen Lenstra, Andries Lenstra, and Jan Karel Lenstra, are also mathematicians. Jan Karel Lenstra is the former director of the Netherlands Centrum Wiskunde & Informatica (CWI). Hendrik Lenstra was the Chairman of the Program Committee of the International Congress of Mathematicians in 2010. Scientific contributions Lenstra has worked principally in computational number theory. He is well known for: * Co-discovering of the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (in 1982); * Developin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjectures That Have Been Proved
In mathematics, a conjecture is a Consequent, conclusion or a proposition that is proffered on a tentative basis without Formal proof, proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of ''Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. Wiles' proof of Fermat's Last Theorem, The first successful proof was released in 1994 by An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
