{"title":"On extended form of the Grothendieck - Serre conjecture","authors":"I. Panin","doi":"10.1070/im9151","DOIUrl":"https://doi.org/10.1070/im9151","url":null,"abstract":"","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58576594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On One Promotion in the Proof of the Hypothesis of the Meromorphic Solutions of the Briot-Bouquet Equations","authors":"Aleksandr Yakovlevich Yanchenko","doi":"10.1070/im9265","DOIUrl":"https://doi.org/10.1070/im9265","url":null,"abstract":"","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58577105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we formulate a general problem of extreme functional interpolation of real-valued functions of one variable (for finite differences, this is the Yanenko–Stechkin–Subbotin problem) in terms of divided differences. The least value of the -th derivative in , , needs to be calculated over the class of functions interpolating any given infinite sequence of real numbers on an arbitrary grid of nodes, infinite in both directions, on the number axis for the class of interpolated sequences for which the sequence of -th order divided differences belongs to . In the present paper this problem is solved in the case when . The indicated value is estimated from above and below using the greatest and the least step of the grid of nodes.
{"title":"Extremal interpolation with the least value of the norm of the second derivative in","authors":"V. T. Shevaldin","doi":"10.1070/IM9125","DOIUrl":"https://doi.org/10.1070/IM9125","url":null,"abstract":"In this paper we formulate a general problem of extreme functional interpolation of real-valued functions of one variable (for finite differences, this is the Yanenko–Stechkin–Subbotin problem) in terms of divided differences. The least value of the -th derivative in , , needs to be calculated over the class of functions interpolating any given infinite sequence of real numbers on an arbitrary grid of nodes, infinite in both directions, on the number axis for the class of interpolated sequences for which the sequence of -th order divided differences belongs to . In the present paper this problem is solved in the case when . The indicated value is estimated from above and below using the greatest and the least step of the grid of nodes.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"86 1","pages":"203 - 219"},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58576418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract. We obtain estimates for integrals of derivatives of rational functions in multiply connected domains in the plane. A sharp order of the growth is found for the integral of the modulus of the derivative of a finite Blaschke product in the unit disk. We also extend the results of E.P. Dolzhenko about the integrals of the derivatives of rational functions to a wider class of domains, namely, to domains bounded by rectifiable curves without zero interior angles, and show the sharpness of the obtained results.
{"title":"Estimates for integrals of derivatives of rational functions in multiply connected domains on the plane","authors":"A. Baranov, I. Kayumov","doi":"10.1070/im9248","DOIUrl":"https://doi.org/10.1070/im9248","url":null,"abstract":"Abstract. We obtain estimates for integrals of derivatives of rational functions in multiply connected domains in the plane. A sharp order of the growth is found for the integral of the modulus of the derivative of a finite Blaschke product in the unit disk. We also extend the results of E.P. Dolzhenko about the integrals of the derivatives of rational functions to a wider class of domains, namely, to domains bounded by rectifiable curves without zero interior angles, and show the sharpness of the obtained results.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41576991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Torelli group of a closed oriented surface <inline-formula><tex-math><?CDATA $S_g$?></tex-math><inline-graphic xlink:href="IZV_85_6_1060ieqn2.gif" xlink:type="simple"></inline-graphic></inline-formula> of genus <inline-formula><tex-math><?CDATA $g$?></tex-math><inline-graphic xlink:href="IZV_85_6_1060ieqn3.gif" xlink:type="simple"></inline-graphic></inline-formula> is the subgroup <inline-formula><tex-math><?CDATA $mathcal{I}_g$?></tex-math><inline-graphic xlink:href="IZV_85_6_1060ieqn4.gif" xlink:type="simple"></inline-graphic></inline-formula> of the mapping class group <inline-formula><tex-math><?CDATA $operatorname{Mod}(S_g)$?></tex-math><inline-graphic xlink:href="IZV_85_6_1060ieqn5.gif" xlink:type="simple"></inline-graphic></inline-formula> consisting of all mapping classes that act trivially on the homology of <inline-formula><tex-math><?CDATA $S_g$?></tex-math><inline-graphic xlink:href="IZV_85_6_1060ieqn2.gif" xlink:type="simple"></inline-graphic></inline-formula>. One of the most intriguing open problems concerning Torelli groups is the question of whether the group <inline-formula><tex-math><?CDATA $mathcal{I}_3$?></tex-math><inline-graphic xlink:href="IZV_85_6_1060ieqn6.gif" xlink:type="simple"></inline-graphic></inline-formula> is finitely presented. A possible approach to this problem relies on the study of the second homology group of <inline-formula><tex-math><?CDATA $mathcal{I}_3$?></tex-math><inline-graphic xlink:href="IZV_85_6_1060ieqn6.gif" xlink:type="simple"></inline-graphic></inline-formula> using the spectral sequence <inline-formula><tex-math><?CDATA $E^r_{p,q}$?></tex-math><inline-graphic xlink:href="IZV_85_6_1060ieqn7.gif" xlink:type="simple"></inline-graphic></inline-formula> for the action of <inline-formula><tex-math><?CDATA $mathcal{I}_3$?></tex-math><inline-graphic xlink:href="IZV_85_6_1060ieqn6.gif" xlink:type="simple"></inline-graphic></inline-formula> on the complex of cycles. In this paper we obtain evidence for the conjecture that <inline-formula><tex-math><?CDATA $H_2(mathcal{I}_3;mathbb{Z})$?></tex-math><inline-graphic xlink:href="IZV_85_6_1060ieqn8.gif" xlink:type="simple"></inline-graphic></inline-formula> is not finitely generated and hence <inline-formula><tex-math><?CDATA $mathcal{I}_3$?></tex-math><inline-graphic xlink:href="IZV_85_6_1060ieqn6.gif" xlink:type="simple"></inline-graphic></inline-formula> is not finitely presented. Namely, we prove that the term <inline-formula><tex-math><?CDATA $E^3_{0,2}$?></tex-math><inline-graphic xlink:href="IZV_85_6_1060ieqn9.gif" xlink:type="simple"></inline-graphic></inline-formula> of the spectral sequence is not finitely generated, that is, the group <inline-formula><tex-math><?CDATA $E^1_{0,2}$?></tex-math><inline-graphic xlink:href="IZV_85_6_1060ieqn10.gif" xlink:type="simple"></inline-graphic></inline-formula> remains infinitely generated after taking quotients by the images of the differentials <inline-formula><tex
{"title":"On a spectral sequence for the action of the Torelli group of genus on the complex of cycles","authors":"A. A. Gaifullin","doi":"10.1070/im9116","DOIUrl":"https://doi.org/10.1070/im9116","url":null,"abstract":"The Torelli group of a closed oriented surface <inline-formula>\u0000<tex-math><?CDATA $S_g$?></tex-math>\u0000<inline-graphic xlink:href=\"IZV_85_6_1060ieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> of genus <inline-formula>\u0000<tex-math><?CDATA $g$?></tex-math>\u0000<inline-graphic xlink:href=\"IZV_85_6_1060ieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> is the subgroup <inline-formula>\u0000<tex-math><?CDATA $mathcal{I}_g$?></tex-math>\u0000<inline-graphic xlink:href=\"IZV_85_6_1060ieqn4.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> of the mapping class group <inline-formula>\u0000<tex-math><?CDATA $operatorname{Mod}(S_g)$?></tex-math>\u0000<inline-graphic xlink:href=\"IZV_85_6_1060ieqn5.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> consisting of all mapping classes that act trivially on the homology of <inline-formula>\u0000<tex-math><?CDATA $S_g$?></tex-math>\u0000<inline-graphic xlink:href=\"IZV_85_6_1060ieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>. One of the most intriguing open problems concerning Torelli groups is the question of whether the group <inline-formula>\u0000<tex-math><?CDATA $mathcal{I}_3$?></tex-math>\u0000<inline-graphic xlink:href=\"IZV_85_6_1060ieqn6.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> is finitely presented. A possible approach to this problem relies on the study of the second homology group of <inline-formula>\u0000<tex-math><?CDATA $mathcal{I}_3$?></tex-math>\u0000<inline-graphic xlink:href=\"IZV_85_6_1060ieqn6.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> using the spectral sequence <inline-formula>\u0000<tex-math><?CDATA $E^r_{p,q}$?></tex-math>\u0000<inline-graphic xlink:href=\"IZV_85_6_1060ieqn7.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> for the action of <inline-formula>\u0000<tex-math><?CDATA $mathcal{I}_3$?></tex-math>\u0000<inline-graphic xlink:href=\"IZV_85_6_1060ieqn6.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> on the complex of cycles. In this paper we obtain evidence for the conjecture that <inline-formula>\u0000<tex-math><?CDATA $H_2(mathcal{I}_3;mathbb{Z})$?></tex-math>\u0000<inline-graphic xlink:href=\"IZV_85_6_1060ieqn8.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> is not finitely generated and hence <inline-formula>\u0000<tex-math><?CDATA $mathcal{I}_3$?></tex-math>\u0000<inline-graphic xlink:href=\"IZV_85_6_1060ieqn6.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> is not finitely presented. Namely, we prove that the term <inline-formula>\u0000<tex-math><?CDATA $E^3_{0,2}$?></tex-math>\u0000<inline-graphic xlink:href=\"IZV_85_6_1060ieqn9.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> of the spectral sequence is not finitely generated, that is, the group <inline-formula>\u0000<tex-math><?CDATA $E^1_{0,2}$?></tex-math>\u0000<inline-graphic xlink:href=\"IZV_85_6_1060ieqn10.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> remains infinitely generated after taking quotients by the images of the differentials <inline-formula>\u0000<tex","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"196 ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138514158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove new results on the derivative of the Minkowski question mark function.
我们证明了Minkowski问号函数导数的新结果。
{"title":"The derivative of the Minkowski function","authors":"D. Gayfulin, I. D. Kan","doi":"10.1070/IM9039","DOIUrl":"https://doi.org/10.1070/IM9039","url":null,"abstract":"We prove new results on the derivative of the Minkowski question mark function.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"85 1","pages":"621 - 665"},"PeriodicalIF":0.8,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43256552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce and describe the Newton polyhedron related to a “minimal” counterexample to the Jacobian conjecture. This description allows us to obtain a sharper estimate for the geometric degree of the polynomial mapping given by a Jacobian pair and to give a new proof in the case of the Abhyankar’s two characteristic pairs.
{"title":"On the Newton polyhedron of a Jacobian pair","authors":"L. Makar-Limanov","doi":"10.1070/IM9067","DOIUrl":"https://doi.org/10.1070/IM9067","url":null,"abstract":"We introduce and describe the Newton polyhedron related to a “minimal” counterexample to the Jacobian conjecture. This description allows us to obtain a sharper estimate for the geometric degree of the polynomial mapping given by a Jacobian pair and to give a new proof in the case of the Abhyankar’s two characteristic pairs.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"85 1","pages":"457 - 467"},"PeriodicalIF":0.8,"publicationDate":"2021-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45631234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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