Koblitz N. (1984) p-adic Numbers, p-adic Analysis, and Zeta Functions. SpringerVerlag, New York, GTM Vol. 58.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= 7 and n 7: e.g. 2 Delta Delta Delta = 4: We remark that 2 = 4 holds, too. Notice that the recurrence relation S(N; K) K Delta S(N Gamma 1; K) S(N Gamma 1; K Gamma 1) implies that 2 Gamma 1) 0 for every sufficiently large n. By the theory of p adic numbers [6] and (12) we can derive that, for all sufficiently large n, 2 k d(k) 2 i=k 1 Gamma k d(k where 2 a=b is defined as 2 (a) Gamma 2 (b) if a and b are integers. This fact helps us to make observations for some special cases. For instance, if n m 3; then ....

Koblitz, N., p-adic numbers, p-adic analysis, and zeta-functions, 2nd edition, Springer-Verlag, New York, 1984

.... in the sequence of moduli p 2 ; p 2 2 ; p 2 3 ; p 2 4 ; I will present in this section a less efficient but more intuitive method is based on the notion of extending p adic numbers in the sequence of moduli p 2 ; p 3 ; p 4 ; p 5 ; as presented by Koblitz [3]. Suppose we have a factorization of f modulo p i , i.e. f(x) j h(x)g(x) mod p i ) 8) where h ffl Z[x] is monic has degree l and reduces to (h mod p) in F p [x] We would like to find polynomials u(x) u 0 u 1 x Delta Delta Delta u l x l and v(x) v 0 v 1 x Delta Delta Delta ....

N. Koblitz. p-adic Numbers, p-adic Analysis, and Zeta Functions. Springer. N. Y., 1984.

....4] Let p 1 ; p 2 ; p n be the distinct prime divisors of f 1 f 2 other than 3. Then f L = ae 9p 1 Delta Delta Delta p n if 3jf 1 f 2 p 1 Delta Delta Delta p n if 36 j f 1 f 2 . In the following section, we will need Hensel s Lemma repeatedly. We state here the version found in [Ko]. Lemma 5 (Hensel s Lemma) Let F (x) 2 Z p and let a 0 2 Z p with F (a 0 ) j 0 (p) and F 0 (a 0 ) j 0 (p) Then there exists a unique a 2 Z p such that F (a) 0 and a j a 0 (p) 2 Determining Local Degrees In this section, we determine conditions under which [L p : Q p ] 9 for a given ....

N. Koblitz. p-adic Numbers, p-adic Analysis, and Zeta-Functions. Springer, New York, 2nd edition, 1984.

....now to show that if k is suciently large, then the polynomial F k (x) is irreducible. For a prime p and an integer a, we de ne (a) p (a) e where p e jja. We de ne the Newton polygon of a polynomial F (x) P n j=0 a j x j as the lower convex hull of the points (j; a j ) cf. 3] [6], 15] We consider the Newton polygon of a polynomial F (x) Let the lattice points along the edges be (x 0 ; y 0 ) x 1 ; y 1 ) x s ; y s ) with 0 = x 0 x 1 x s = deg F (x) Then the degree of any irreducible factor of F (x) over Z[x] must be some sum of the di erences ....

....polygon is de ned as the lower convex hull of the points (j; a j ) Throughout the remainder of this paper, we work in an algebraic closure of Q p unless noted otherwise or unless it is clear from the context that we are working in C . As references, we mention the books of Gouv ea [3] Koblitz [6], and Weis [15] 9 A lemma we will make use of throughout the remainder of the paper is the following. Lemma 6. Let be an m th p adic root of unity and 0 an m 0 th p adic root of unity with 0 6= Suppose p mm 0 . Then ( 0 ) 0. The lemma follows from Lemma 2.12 of ....

N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, Springer-Verlag, New York, 1977.

....computation, its only advantage is that it knows the polynomial f(x) explicitly. Here we design a polynomial time algorithms of zero testing of polynomials of class P p (m; n) by using a black box of the aforementioned type. Sparse polynomials are considered as well. Using the Strassman theorem [9] one can apply our result to zero testing of various analytic functions over p adic fields, exponential polynomials of the form E(X) r X i=1 f i (X) g i (X) i ; 2) where i 2 C p , f i (X) 2 C p [X] g i (X) 2 ZZ[X] in particular. The we consider polynomials (1) with coefficients from ....

N. Koblitz, p-adic numbers, p-adic analysis, and zeta-functions, Springer-Verlag, Berlin, 1977.

....now to show that if k is sufficiently large, then the polynomial F k (x) is irreducible. For a prime p and an integer a, we define (a) p (a) e where p e jja. We define the Newton polygon of a polynomial F (x) P n j=0 a j x j as the lower convex hull of the points (j; a j ) cf. 3] [6], 15] We consider the Newton polygon of a polynomial F (x) Let the lattice points along the edges be (x 0 ; y 0 ) x 1 ; y 1 ) x s ; y s ) with 0 = x 0 x 1 Delta Delta Delta x s = deg F (x) Then the degree of any irreducible factor of F (x) over Z[x] must be some sum of ....

....polygon is defined as the lower convex hull of the points (j; a j ) Throughout the remainder of this paper, we work in an algebraic closure of Q p unless noted otherwise or unless it is clear from the context that we are working in C . As references, we mention the books of Gouvea [3] Koblitz [6], and Weis [15] A lemma we will make use of throughout the remainder of the paper is the following. Lemma 6. Let i be an m th p adic root of unity and i 0 and m 0 th p adic root of unity. Suppose p mm 0 . Then (i Gamma i 0 ) 0. The lemma follows from Lemma 2.12 of [14] It is also ....

N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, Springer-Verlag, New York, 1977.

....(1 2i) and p 2 = 1 Gamma 2i) are prime ideals. We have j 1 2i j p 1 = 1=5 while j 1 Gamma 2i j p 1 = 1, and similarly j 1 2i j p 2 = 1 while j 1 Gamma 2i j p 2 = 1=5. Note that the p 1 and p 2 adifications of Z[i] are just Z 5 since Z 5 contains a square root of Gamma1 (cf. e.g. [16]) The fiber of the solenoid S over 0 is Z 2 5 . There are two non Archimedean Lyapunov exponents 1 and 2 , each with multiplicity 1 given by 1 (x; y) Gammax log 5 and 2 (x; y) Gammay log 5 where (x; y) 2 Z 2 . Also note that there is one Archimedean Lyapunov exponent of multiplicity ....

N. Koblitz, p-adic numbers, p-adic analysis, and zeta functions, Springer, New York 1984.

....by p sm . By (7.5) it follows that for any r # N which is prime to p (and hence invertible in A) the number of r th roots of unity in A is # p sm . For p #= 2, Q p has no p th roots of unity other than 1 (because then p is unramified in Q p but totally ramified in Q p ( p ) cf. [23] p. 20 exercise 14) For p = 2, Q p contains no square root of 1. For any field M , let M # denote its subfield generated by x # M : x ( p n ) 1 for some n # N . Then Q # p = Q . Hence [L # 0 : Q ] # sm. Hence x # L 0 : x ( p n ) 1 for some n # N # 2sm. Hence ....

Koblitz, N.: p-adic Numbers, p-adic Analysis, and Zeta-Functions, Springer-Verlag (New York), 1977.

....and Q p the completion of Q with respect to the p adic valuation j j p de ned on Q by (1.1) j0j p = 0 and jAj p = p if A = p r=s, where p j =r; p j =s. Then Q p is the eld of p adic numbers with p adic valuation j j p , the extension of the original valuation on Q (cf. Koblitz [12] or Schikhof [19] It is well known that every A 2 Q p has a unique series representation A = n=v(A) c n p , c n 2 f0; 1; 2; p 1g. In the discussion below we call the nite series hAi = P v(A) n 0 c n p the fractional part of A. Then hAi 2 S p , where we de ne S p = fhAi : A 2 ....

Koblitz N, P -adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd Ed., Springer, 1984.

....for the p adic order of the digits and the p adic order of approximation by the partial sums of the series expansions. 1. Introduction Let Q be the eld of rational numbers, p a prime number and Q p the completion of Q with respect to the p adic absolute value j j p de ned on Q by (cf. Koblitz [8] or Schikhof [12] 1.1) j0j p = 0 and jAj p = p a if A = p a r s ; where p 6 jrs: The exponent a in this de nition is the p adic valuation of A, which we denote by v p (A) It is well known that every A 2 Q p has a unique series representation n=vp (A) c n p , c n 2 f0; 1; 2; p ....

N. Koblitz. p-Adic Numbers, p-Adic Analysis, and Zeta-Functions. SpringerVerlag, New York, 1984.

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Koblitz N. (1984) p-adic Numbers, p-adic Analysis, and Zeta Functions. SpringerVerlag, New York, GTM Vol. 58.

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Neal Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions , SpringerVerlag.

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N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta Functions, Springer-Verlag, 1984.

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Neal Koblitz. p-adic Numbers, p-adic Analysis, and Zeta-Functions. Springer-Verlag, New York, 1977; MR 57#5964.

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N. Koblitz. p-adic Numbers, p-adic Analysis, and Zeta-Functions. Springer-Verlag, New York, 1977.

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N. Koblitz (1977), p-adic numbers, p-adic analysis, and zeta-functions (second edition), Springer-Verlag.

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N. Koblitz (1977), p-adic numbers, p-adic analysis, and zeta-functions (second edition), Springer-Verlag.

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