{"title":"Products of consecutive values of some quartic polynomials","authors":"A. Dubickas","doi":"10.7169/FACM/1733","DOIUrl":"https://doi.org/10.7169/FACM/1733","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.7169/FACM/1733","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46694784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Lucas non-Wieferich primes in arithmetic progressions","authors":"S. S. Rout","doi":"10.7169/facm/1709","DOIUrl":"https://doi.org/10.7169/facm/1709","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.7169/facm/1709","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42696359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that for most primes $p$, the set of Fibonomials forms an additive basis of order $8$ for the group of residue classes modulo $p$.
我们证明了对于大多数素数$p$，对于模$p$的残差类组，纤维组元形成$8$阶的加性基。
{"title":"A note on fibonomial coefficients","authors":"V. C. García, F. Luca","doi":"10.7169/FACM/1697","DOIUrl":"https://doi.org/10.7169/FACM/1697","url":null,"abstract":"We show that for most primes $p$, the set of Fibonomials forms an additive basis of order $8$ for the group of residue classes modulo $p$.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.7169/FACM/1697","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41879877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a recent paper of Dokchitser--Dokchitser--Maistret--Morgan, the authors introduced the concept of a cluster picture associated to a hyperelliptic curve from which they are able to recover numerous invariants, including the inertia representation on the first etale cohomology group of the curve. The purpose of this paper is to explore the functionality of these cluster pictures and prove that the inertia representation of a hyperelliptic curve is a function of its cluster picture.
{"title":"Clusters, inertia, and root numbers","authors":"Matthew Bisatt","doi":"10.7169/facm/1973","DOIUrl":"https://doi.org/10.7169/facm/1973","url":null,"abstract":"In a recent paper of Dokchitser--Dokchitser--Maistret--Morgan, the authors introduced the concept of a cluster picture associated to a hyperelliptic curve from which they are able to recover numerous invariants, including the inertia representation on the first etale cohomology group of the curve. The purpose of this paper is to explore the functionality of these cluster pictures and prove that the inertia representation of a hyperelliptic curve is a function of its cluster picture.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43739262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We bound Kloosterman-like sums of the shape [ sum_{n=1}^N exp(2pi i (x lfloor f(n)rfloor+ y lfloor f(n)rfloor^{-1})/p), ] with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition on $f''$, and $lfloor f(n)rfloor^{-1}$ is meaning inversion modulo~$p$. As an immediate application, we obtain results concerning the distribution of modular inverses inverses $lfloor f(n)rfloor^{-1} pmod{p}$. The results apply, in particular, to Piatetski-Shapiro sequences $ lfloor t^crfloor$ with $cin(1,frac{4}{3})$. The proof is an adaptation of an argument used by Banks and the first named author in a series of papers from 2006 to 2009.
{"title":"Kloosterman sums with twice-differentiable functions","authors":"I. Shparlinski, Marc Technau","doi":"10.7169/facm/1845","DOIUrl":"https://doi.org/10.7169/facm/1845","url":null,"abstract":"We bound Kloosterman-like sums of the shape [ sum_{n=1}^N exp(2pi i (x lfloor f(n)rfloor+ y lfloor f(n)rfloor^{-1})/p), ] with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition on $f''$, and $lfloor f(n)rfloor^{-1}$ is meaning inversion modulo~$p$. As an immediate application, we obtain results concerning the distribution of modular inverses inverses $lfloor f(n)rfloor^{-1} pmod{p}$. The results apply, in particular, to Piatetski-Shapiro sequences $ lfloor t^crfloor$ with $cin(1,frac{4}{3})$. The proof is an adaptation of an argument used by Banks and the first named author in a series of papers from 2006 to 2009.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41520636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent $s d$ case, fixing all pairs of distnaces leads to an overdetermined system, so $q^{binom{k+1}{2}}$ is no longer the correct number of congruence classes. We determine the correct number, and prove that $|E|gtrsim q^s$ still determines a positive proportion of all congruence classes, for the same $s$ as in the $kleq d$ case.
{"title":"Congruence classes of large configurations in vector spaces over finite fields","authors":"Alex McDonald","doi":"10.7169/facm/1814","DOIUrl":"https://doi.org/10.7169/facm/1814","url":null,"abstract":"Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent $s d$ case, fixing all pairs of distnaces leads to an overdetermined system, so $q^{binom{k+1}{2}}$ is no longer the correct number of congruence classes. We determine the correct number, and prove that $|E|gtrsim q^s$ still determines a positive proportion of all congruence classes, for the same $s$ as in the $kleq d$ case.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47520377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For integers $a_0,a_1,ldots,a_n$ with $|a_0a_n|=1$ and either $alpha =u$ with $1leq u leq 50$ or $alpha=u+ frac{1}{2}$ with $1 leq u leq 45$, we prove that $psi_n^{(alpha)}(x;a_0,a_1,cdots,a_n)$ is irreducible except for an explicit finite set of pairs $(u,n)$. Furthermore all the exceptions other than $n=2^{12},alpha=89/2$ are necessary. The above result with $0leqalpha leq 10$ is due to Filaseta, Finch and Leidy and with $alpha in {-1/2,1/2}$ due to Schur.
对于整数$a_0,a_1,ldots,a_n$与$|a_0a_n|=1$, $alpha =u$与$1leq u leq 50$或$alpha=u+ frac{1}{2}$与$1 leq u leq 45$，我们证明了$psi_n^{(alpha)}(x;a_0,a_1,cdots,a_n)$除了一个显式有限对集$(u,n)$外是不可约的。此外，除了$n=2^{12},alpha=89/2$之外的所有例外都是必要的。上面的结果与$0leqalpha leq 10$是由于Filaseta、Finch和Leidy，与$alpha in {-1/2,1/2}$是由于Schur。
{"title":"Irreducibility of extensions of Laguerre polynomials","authors":"S. Laishram, Saranya G. Nair, T. Shorey","doi":"10.7169/facm/1748","DOIUrl":"https://doi.org/10.7169/facm/1748","url":null,"abstract":"For integers $a_0,a_1,ldots,a_n$ with $|a_0a_n|=1$ and either $alpha =u$ with $1leq u leq 50$ or $alpha=u+ frac{1}{2}$ with $1 leq u leq 45$, we prove that $psi_n^{(alpha)}(x;a_0,a_1,cdots,a_n)$ is irreducible except for an explicit finite set of pairs $(u,n)$. Furthermore all the exceptions other than $n=2^{12},alpha=89/2$ are necessary. The above result with $0leqalpha leq 10$ is due to Filaseta, Finch and Leidy and with $alpha in {-1/2,1/2}$ due to Schur.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48545905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Conical measures and closed vector measures","authors":"S. Okada, W. Ricker","doi":"10.7169/FACM/1711","DOIUrl":"https://doi.org/10.7169/FACM/1711","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41764774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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