{"title":"Elementary methods in the study of the Deuring–Heilbronn phenomenon","authors":"C. Bellotti, G. Puglisi","doi":"10.4064/aa220909-23-5","DOIUrl":"https://doi.org/10.4064/aa220909-23-5","url":null,"abstract":"","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70440651","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}
This paper deals with properties of the algebraic variety defined as the set of zeros of a"deficient"sequence of multivariate polynomials. We consider two types of varieties: ideal-theoretic complete intersections and absolutely irreducible varieties. For these types, we establish improved bounds on the dimension of the set of deficient systems of each type over an arbitrary field. On the other hand, we establish improved upper bounds on the number of systems of each type over a finite field.
{"title":"The distribution of defective multivariate polynomial systems over a finite field","authors":"Nardo Giménez, Guillermo Matera, Mariana Pérez, Melina Privitelli","doi":"10.4064/aa220817-21-7","DOIUrl":"https://doi.org/10.4064/aa220817-21-7","url":null,"abstract":"This paper deals with properties of the algebraic variety defined as the set of zeros of a\"deficient\"sequence of multivariate polynomials. We consider two types of varieties: ideal-theoretic complete intersections and absolutely irreducible varieties. For these types, we establish improved bounds on the dimension of the set of deficient systems of each type over an arbitrary field. On the other hand, we establish improved upper bounds on the number of systems of each type over a finite field.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"133 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135317987","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}
Let $tau (n)$ denote the number of positive divisors of an integer $ngeq 1$ and let $lfloor cdot rfloor $ denote the integer part function. We evaluate asymptotically the sums $$ sum _{nleq x}f (lfloor x/nrfloor )tau (n), $$ where $f$ is an arit
{"title":"Hyperbolic summation for fractional sums","authors":"Meselem Karras, Ling Li, Joshua Stucky","doi":"10.4064/aa230331-31-7","DOIUrl":"https://doi.org/10.4064/aa230331-31-7","url":null,"abstract":"Let $tau (n)$ denote the number of positive divisors of an integer $ngeq 1$ and let $lfloor cdot rfloor $ denote the integer part function. We evaluate asymptotically the sums $$ sum _{nleq x}f (lfloor x/nrfloor )tau (n), $$ where $f$ is an arit","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135400230","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":"Covering systems with large moduli associated with reducible shifts of integer polynomials","authors":"Pradipto Banerjee","doi":"10.4064/aa220518-18-3","DOIUrl":"https://doi.org/10.4064/aa220518-18-3","url":null,"abstract":"","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70440111","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}
Let $kge 2$ be a positive integer and $P^+(n)$ the greatest prime factor of a positive integer $n$ with convention $P^+(1)=1$. For any $theta in left [frac 1{2k},frac {17}{32k}right )$, set $$T_{k,theta }(x)=sum _{substack {p_1cdot cdot c
{"title":"Solution to a problem of Luca, Menares and Pizarro-Madariaga","authors":"Yuchen Ding, Lilu Zhao","doi":"10.4064/aa230604-12-7","DOIUrl":"https://doi.org/10.4064/aa230604-12-7","url":null,"abstract":"Let $kge 2$ be a positive integer and $P^+(n)$ the greatest prime factor of a positive integer $n$ with convention $P^+(1)=1$. For any $theta in left [frac 1{2k},frac {17}{32k}right )$, set $$T_{k,theta }(x)=sum _{substack {p_1cdot cdot c","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135400210","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}
Let $e_1,e_2,e_3$ be nonzero integers satisfying $e_1+e_2+e_3=0$. Let $(a,b,c)$ be a primitive triple of odd integers satisfying $e_1a^2+e_2b^2+e_3c^2=0$. Denote by $E: y^2=x(x-e_1)(x+e_2)$ and $mathcal E: y^2=x(x-e_1a^2)(x+e_2b^2)$. Assume that the $2$-Selmer groups of $E$ and $mathcal E$ are minimal. Let $n$ be a positive square-free odd integer, where the prime factors of $n$ are nonzero quadratic residues modulo each odd prime factor of $e_1e_2e_3abc$. Then under certain conditions, the $2$-Selmer group and the Cassels pairing of the quadratic twist $E^{(n)}$ coincide with those of $mathcal E^{(n)}$. As a corollary, $E^{(n)}$ has Mordell-Weil rank zero without order $4$ element in its Shafarevich-Tate group, if and only if these holds for $mathcal E^{(n)}$. We also give some applications for the congruent elliptic curve.
{"title":"On a comparison of Cassels pairings of different elliptic curves","authors":"Shenxing Zhang","doi":"10.4064/aa220709-15-7","DOIUrl":"https://doi.org/10.4064/aa220709-15-7","url":null,"abstract":"Let $e_1,e_2,e_3$ be nonzero integers satisfying $e_1+e_2+e_3=0$. Let $(a,b,c)$ be a primitive triple of odd integers satisfying $e_1a^2+e_2b^2+e_3c^2=0$. Denote by $E: y^2=x(x-e_1)(x+e_2)$ and $mathcal E: y^2=x(x-e_1a^2)(x+e_2b^2)$. Assume that the $2$-Selmer groups of $E$ and $mathcal E$ are minimal. Let $n$ be a positive square-free odd integer, where the prime factors of $n$ are nonzero quadratic residues modulo each odd prime factor of $e_1e_2e_3abc$. Then under certain conditions, the $2$-Selmer group and the Cassels pairing of the quadratic twist $E^{(n)}$ coincide with those of $mathcal E^{(n)}$. As a corollary, $E^{(n)}$ has Mordell-Weil rank zero without order $4$ element in its Shafarevich-Tate group, if and only if these holds for $mathcal E^{(n)}$. We also give some applications for the congruent elliptic curve.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136260016","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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