A. Borisov, On some polynomials allegedly related to the abc conjecture. Acta Arith. 84 (1998), 109-128. Home/Search Document Details and Download
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An ABC construction of number fields - Roberts (Correct)
....of the monodromy transformation m c in G. Then Gal(K c =Q) is contained in Norm c =hm c i. For the covers XN;m of this section, the most interesting cusp to specialize at is c = 1. The Galois group GN;m;1 is contained in a symmetric group SN 2 . These elds KN;m;1 C have been studied in [Bor]. In our example, one has f 9;1 (1; x) x 1) 7x 8) The septic eld de ned by the degree seven factor has Galois group S 7 and discriminant 2 . In general, cuspidal specializations seem worthy of special attention. First, they depend only on the group theoretic data ....
A. Borisov, On some polynomials allegedly related to the abc conjecture. Acta Arith. 84 (1998), 109-128.
Septic Fields With Discriminant - Jones, Roberts (Correct)
....4 5x 3 6x 2 7x 8) The septic factor has polynomial discriminant 2 16 3 12 and the septic eld that it de nes is Field 10 from the previous section. One couldn t ask for more simplybehaved coecients, and this septic polynomial is an example of an abc polynomial, in the sense of [Bor]. The polynomial f t (x) in (4.1) can be regarded as a family of separable nonic polynomials in Q[x] indexed by t 2 Q f0; 1g. One sees from (4.2) that for some of these t, the discriminant of f t (x) has the form 2 a 3 b . For other t, the polynomial discriminant will have an extra square ....
Borisov, A., On some polynomials allegedly related to the abc conjecture, Acta. Arith. 84 (1998), no. 2, 109-128
Septic Fields With Discriminant - Jones, Roberts (Correct)
....4 5x 3 6x 2 7x 8) The septic factor has polynomial discriminant 2 16 3 12 and the septic eld that it de nes is Field 10 from the previous section. One couldn t ask for more simplybehaved coecients, and this septic polynomial is an example of an abc polynomial, in the sense of [Bor]. The polynomial f t (x) in (4.1) can be regarded as a family of separable nonic polynomials in Q[x] indexed by t 2 Q f0; 1g. One sees from (4.2) that for some 8 JOHN W. JONES AND DAVID P. ROBERTS of these t, the discriminant of f t (x) has the form 2 a 3 b . For other t, the polynomial ....
Borisov, A., On some polynomials allegedly related to the abc conjecture, Acta. Arith. 84 (1998), no. 2, 109-128
Classes Of Polynomials Having Only One Non-Cyclotomic.. - Borisov Filaseta Lam (1998) Self-citation (Borisov) (Correct)
....n . Then f 0 (x) is irreducible over the rationals. He noted then that the conjecture is true if n = p 1 2 or if n = p r where p is a prime and r a positive integer. Calculations showed the conjecture also held for n 100. Recently, in a study of more general polynomials, the rst author [2] obtained further irreducibility results for f(x) in particular, he established irreducibility in the case that n 1 is a squarefree number 3 and in the case that n = 2p 1 where p is prime. The third author independently observed that f (k) x) is Eisenstein if n = p 1 for every integer k ....
....0 ) 1 1 is a root of P mm 0 1 j=0 (x 1) j , a monic polynomial with constant term relatively prime to p. We will make particular use of the lemma with 0 = 1. The next result, an essential ingredient to our arguments for Theorems 1 and 2, is based on the work of the rst author in [2]. Proposition 1. Let w(x) P n 1 j=0 a j x j 2 Z[x] with a n 1 6= 0, and let m and r be integers with m 0, r 0, n 1 = m r. Let p be a prime such that pjm, p r, and p a n 1 . Write m = p m 0 where p (m 0 ) 0. Suppose that w(x) a n 1 (x m 1)x r (mod p ) and ....
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A. Borisov, On some polynomials allegedly related to the abc conjecture, Acta Arith. 84 (1998), 109-128.
Statement on Research - Borisov Self-citation (Borisov) (Correct)
....to Number Theory (cf. 6] 12] 2. Fano varieties with log terminal singularities (cf. 5] 10] 13] 3. Convex lattice polytopes and cones with few lattice points inside (cf. 7] 8] 13] 11] citeterminal, 15] 4. Distribution of roots and irreducibility of polynomials (cf. 9] 16] Let me now describe in some more details my contributions and research plans in the above areas. I want to note specifically that the third topic is very well suited for the research involving undergraduates, and I already used it for this purpose. 1. Applications of Harmonic Analysis ....
....to Number Theory The ultimate goal of this part of my research is to enrich the Arakelov theory by providing a conceptual framework for the global spaces and cohomologies of hermitian vector bundles. So far I was able to do it in dimension one, developing ideas from the Tate s thesis (cf. 39] and a preprint by van der Geer and Schoof (cf. 19] Generally speaking, the main method of this part of my research is to explore deep similarities between Number Theory and Algebraic Geometry, using Harmonic Analysis structures to define arithmetic analogs of geometric objects. More ....
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A. Borisov. On some polynomials allegedly related to the abc conjecture, Acta Arithmetica, 84 (1998), no. 2, 109--128.
Classes Of Polynomials Having Only One.. - Borisov, Filaseta.. (1998) Self-citation (Borisov) (Correct)
....f 0 (x) is irreducible over the rationals. He noted then that the conjecture is true if n = p Gamma 1 2 or if n = p r where p is a prime and r a positive integer. Calculations showed the conjecture also held for n 100. Recently, in a study of more general polynomials, the first author [2] obtained further irreducibility results for f(x) in particular, he established irreducibility in the case that n 1 is a squarefree number 3 and in the case that n = 2p Gamma 1 where p is prime. The third author independently observed that f (k) x) is Eisenstein if n = p Gamma 1 for ....
....is a root of P mm 0 Gamma1 j=0 (x 1) j , a monic polynomial with constant term relatively prime to p. We will make particular use of the lemma with i 0 = Sigma1. The next result, an essential ingredient to our arguments for Theorems 1 and 2, is based on the work of the first author in [2]. Proposition 1. Let w(x) P n 1 j=0 a j x j 2 Z[x] with a n 1 6= 0, and let m and r be integers with m 0, r 0, n 1 = m r. Let p be a prime such that pjm, p r, and p a n 1 . Write m = p m 0 where p (m 0 ) 0. Suppose that w(x) j a n 1 (x m Gamma 1)x r (mod p ) ....
[Article contains additional citation context not shown here]
A. Borisov, On some polynomials allegedly related to the abc conjecture, Acta Arith. 84 (1998), 109--128.
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