Leírás
Győry Kálmán online előadást tart június 27-én csütörtökön 14:00 órakor.
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Kálmán Győry will give an online talk on Thursday, the 27th of June at 2 pm.
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Kálmán Győry, General effective reduction theory of integral polynomials and its applications (survey with a brief historical overview)
Abstract : The reduction theory of integral polynomials with non-zero discriminant was initiated by Lagrange (1773 ). For quadratic polynomials he proved that up to the classical GL_2( Z ) -equivalence resp. Z-equivalence ( monic case ) there are only finitely many such polynomials. This was extended by Hermite (1951) to cubic polynomials. Hermite (1857) attempted to extend these results to the general case, to polynomials of any given degree n > 3. He introduced a new notion of equivalence and proved in an ineffective way a finiteness result on the corresponding equivalence classes of integral polynomials of degree n. But he did not compare his equivalence with the above-mentioned classical equivalences. Hermite (1857) result does not appear to have been studied in the literature until Narkiewicz (2018) excellent book , where Hermite equivalence was mixed up with the classical equivalences. In our recent paper BEGyRS (2023) with Bhargava, Evertse , Remete and Swaminathan we provided a thorough treatment of the notion of Hermite equivalence, and proved that the classical equivalences are much more precise than Hermite equivalence. As a consequence, in BEGyRS (2023) it turned out that Hermite original objective - proving that there are only finitely many GL_2( Z ) - resp. Z-equivalence classes of integral polynomials of given degree and given non-zero discriminant - was finally achieved more than a century later by Birch and Merriman (1972) and independently, for monic polynomials and in a more precise and effective form by Győry (1973). The result of Birch and Merriman was subsequently made effective by Evertse and Győry (1991). In other words, Győry (1973) and Evertse and Győry (1991) together solved the above-mentioned old
problem of Hermite (1857 ) affirmatively, in an effective way, which resulted in many significant consequences and applications. For example, in the 1970's I provided as a consequence of my Győry (1973) paper the first general effective finiteness theorems for the monogeneity and power integral bases of number fields. For later various applications and generalizations we refer to the monograph of Evertse and Győry, Discriminant equations in Diophantine number theory, Cambridge, 2017.
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Andras Biro is inviting you to a scheduled Zoom meeting.
Topic: szeminarium
Time: Jun 27, 2024 02:00 PM Budapest
Join Zoom Meeting
https://us06web.zoom.us/j/87235726422?pwd=it3ZZh5LRuR3J3WkiuVa8m2oB0cRGC.1
Meeting ID: 872 3572 6422
Passcode: 602919