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Timestamp: 2019-04-24 11:08:27+00:00

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PhD Phis. -Math. Sci. Kovalevskaya E.I., PhD Phis. -Math. Sci.
1. We extend the author’ earlier result  to inhomogeneous case.
Q = R´C´ and Q1 = R´ (k ³ 2). We define a measure in Q as a product of the Lebesque measures m and m¢ in R and R2 respectively, and the Haar measure in (1 £ i £ k), that is, .We define also a measure in Θ1 as a product of the Lebesque measure m in R and the Haar measure in , that is, .
where Θ and Θ. The parameters satisfy the following conditions: max min , Besides, we have the conditions and in the case , and in the case . Note that the first conditions for the parameters except the trivial inequalities in (2) and (3). We prove two theorems.
Theorem 1. For every vector Θ1 the system of inequalities (2) has only a finite number of solutions in polynomials P2 for almost all Θ1.
Theorem 2. For every vector Θ the system of inequalities (3) has only a finite number of solutions in polynomials P3 for almost all Θ.
Other words, we can say that according to the theorem 1 for every vector Θ1 the system of inequalities (2) has infinitely many solutions in polynomials P2 for almost none Θ1. Similarly we can get a new formulation of the theorem 2. The condition (1) is a crucial condition for such a metric characteristic. If the opposite condition carries out, i. e., then according to the Khintcine theorem (1924) we can expect that for every vector Θ1 the system of inequalities (2) has infinitely many solutions in polynomials P2 for almost all Θ1 and we can have a similarly result for (3). Thus, we see that these metric assertions have the character of assertions of the “almost none” or “almost all” type (so-called zero-one law). Hence, the problem under consideration belongs to the metric theory of Diophantine approximation on manifolds which is developed intensively this time [1-4].
The situation is different when we consider instead of where d is an irrational number. With a point of view of the continuity it is readily proved that for any we can f ind such a number d that where p1 and p2 are the zeros of . Besides, the zeros of are the transcendental numbers if d is the same number.
Note that the inhomogeneous Diophantine approximation for the Veronese curve were investigated earlier by V. Bernik, H. Dickinson and M. Dodson  in only, V. Bernik, H. Dickinson and J. Yuan  in only.
In order to prove the theorems we need four lemmas. As in [5, p. 32, 93], the investigation of systems (2) and (3) can be reduced to the case of the primitive irreducible polynomials Pn with and . We denote the set of those polynomials as be the subset of polynomials for which and H is a sufficient large integer. Let be the zeros of the polynomial Pn in C and be the zeros of the polynomial Pn in , where be the smallest field containing and all algebraic numbers, .
For proof see Lemma 7 [5, p.19].
Definition. We denote the smallest m for which (4) is true by m0.
Lemma 2. Let and with where c is a constant depending only on n, . Then for every root of Pn .
For proof see Lemma 1 [5, p.13].
Lemma 3. Let and with where c1 is a constant depending only on n. Then for every root of Pn .
For proof see Lemma 3 .
be the two systems of inequalities with Θ1 and Θ respectively, where H, Θ and parameters are defined as in the theorems 2, 3. Then the systems of inequalities (5) and (6) are satisfied by at most finitely many polynomials and for almost all Θ1 and for almost all Θ respectively.
This is a main theorem in .
2. Reduction to a polynomial and Proof the theorems.
In the next we use the following notation: 1) [X], the integral part of ,2) is equivalent to the simultaneous validity of and .
where N depends on the height of the polynomial Pn associated with , i.e., .
where and. Note that as above we may assume without loss of generality that if . Hence, we have N if the height is sufficient large and .
Now according to the lemma 4 the theorems 1,2 are proved.
ACKNOWLEDGMENT. The research was done in the limits of the Belorussian State Programme of Fundamental researches (Project 05-K-065).
1. BERNIK V.I. – DICKINSON H. – DODSON M.: Approximation of real numbers by integral polynomials, Dokl. Nats. Akad. Nayk Belarusi 42 (1997), 51-54.
2. BERNIK V. – DICKINSON H. – YUAN J.: inhomogeneous diophantine approximation on polynomials, Acta Arith. 90 (1999), 37-48.
3. BERESNEVICH V. – BERNIK V. – KLEINBOCK D. – MARGULIES G. A.: Metric Diophantine approximation: the Khintchine-Groshev theorem for non-degenerate manifolds, Moscow Math. J. 2 (2002), 203-225.
4. KOVALEVSKAYA E. I. – MOROZOVA I. M.: Zero-one law in Diophantine approximation for the points of Veronese’s curve of the second and the third degrees with respect to different valuations, Materially III mizhnarod. nauk.-pract. conf. “Aktualni problemi suchsnykh nauk: teoriya ta practica – 2006”. 16-30 chervnya 2006 roku. 22. Tekhichni nauki. Dnipropetrovsk. Nauki I osvita (2006), 14-17.
5. SPRINDZUK V. G.: Mahler’s problem in metric Number Theory, Nauka i Tehnika, Minsk, 1967 (Russian).
6. BAKER A.: On the theorem of Sprindzuk, Proc. Roy. Soc. London Ser. A 242 (1966), 94-104.

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