The Experts below are selected from a list of 10788 Experts worldwide ranked by ideXlab platform
Ronald J. Evans - One of the best experts on this subject based on the ideXlab platform.
-
some mixed Character Sum identities of katz
Journal of Number Theory, 2017Co-Authors: Ronald J. EvansAbstract:Abstract A conjecture connected with quantum physics led N. Katz to discover some amazing mixed Character Sum identities over a field of q elements, where q is a power of a prime p > 3 . His proof required deep algebro-geometric techniques, and he expressed interest in finding a more straightforward direct proof. Such a proof has been given by Evans and Greene in the case q ≡ 3 ( mod 4 ) , and in this paper we give a proof for the remaining case q ≡ 1 ( mod 4 ) . Moreover, we show that the identities are valid for all Characteristics p > 2 .
-
Some mixed Character Sum identities of Katz II
Research in Number Theory, 2017Co-Authors: Ronald J. Evans, John R GreeneAbstract:A conjecture connected with quantum physics led N. Katz to discover some amazing mixed Character Sum identities over a field of q elements, where q is a power of a prime \(p >3\). His proof required deep algebro-geometric techniques, and he expressed interest in finding a more straightforward direct proof. The first author recently gave such a proof of his identities when \(q \equiv 1 \pmod 4\), and this paper provides such a proof for the remaining case \(q \equiv 3 \pmod 4\). Our proofs are valid for all Characteristics \(p>2\). Along the way we prove some elegant new Character Sum identities.
-
Some mixed Character Sum identities of Katz
arXiv: Number Theory, 2016Co-Authors: Ronald J. EvansAbstract:A conjecture connected with quantum physics led N. Katz to discover some amazing mixed Character Sum identities over a field of q elements, where q is a power of a prime p > 3. His proof required deep algebro-geometric techniques, and he expressed interest in finding a more straightforward direct proof. The first author recently gave such a proof of his identities when q = 1 (mod 4), and this paper provides such a proof for the remaining case q = 3 (mod 4). Our proofs are valid for all Characteristics p > 2. Along the way we prove some elegant new Character Sum identities.
-
Selberg-Jack Character Sums of Dimension 2
Journal of Number Theory, 1995Co-Authors: Ronald J. EvansAbstract:Abstract Kadell extended Selberg′s n-dimensional beta integral formula by inserting a normalized Jack polynomial as a factor in the integrand. We formulate and prove a Character Sum analog of Kadell′s formula in the case n = 2, raising hope that such an analog may exist for general n.
-
A Character Sum for root system
Proceedings of the American Mathematical Society, 1992Co-Authors: Ronald J. EvansAbstract:A Character Sum analog of the Macdonald-Morris constant term identity for the root system G2 is proved. The proof is based on recent evaluations of Selberg Character Sums and on a Character Sum analog of Dixon's Summation formula. A conjectural evaluation is presented for a related Sum.
Avi Wigderson - One of the best experts on this subject based on the ideXlab platform.
-
Extractors And Rank Extractors For Polynomial Sources
Computational Complexity, 2009Co-Authors: Zeev Dvir, Ariel Gabizon, Avi WigdersonAbstract:In this paper we construct explicit deterministic extractors from polynomial sources, which are distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources (which are degree 1 polynomials). A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields. The steps in our extractor construction, and the tools (mainly from algebraic geometry) that we use for them, are of independent interest: The first step is a construction of rank extractors, which are polynomial mappings which ‘extract’ the algebraic rank from any system of low degree polynomials. More precisely, for any n polynomials, k of which are algebraically independent, a rank extractor outputs k algebraically independent polynomials of slightly higher degree. The rank extractors we construct are applicable not only over finite fields but also over fields of Characteristic zero. The next step is relating algebraic independence to min-entropy. We use a theorem of Wooley to show that these parameters are tightly connected. This allows replacing the algebraic asSumption on the source (above) by the natural information theoretic one. It also shows that a rank extractor is already a high quality condenser for polynomial sources over polynomially large fields. Finally, to turn the condensers into extractors, we employ a theorem of Bombieri, giving a Character Sum estimate for polynomials defined over curves. It allows extracting all the randomness (up to a multiplicative constant) from polynomial sources over exponentially large prime fields.
-
FOCS - Extractors and Rank Extractors for Polynomial Sources
48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07), 2007Co-Authors: Zeev Dvir, Ariel Gabizon, Avi WigdersonAbstract:In this paper we construct explicit deterministic extractors from polynomial sources, namely from distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources. A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields. The first step towards extraction is a construction o/rank extractors, which are polynomial mappings that "extract" the algebraic rank from any system of low degree polynomials. More precisely, for any n polynomials, k of which are algebraically independent, a rank extractor outputs k algebraically independent polynomials of slightly higher degree. A result of Wooley allows us to relate algebraic rank and min-entropy and to show that a rank extractor is also a high quality condenser for polynomial sources over polynomially large fields. Finally, to turn this condenser into an extractor, we employ a theorem of Bombieri, giving a Character Sum estimate for polynomials defined over curves. It allows extracting all the randomness (up to a multiplicative constant) from polynomial sources over exponentially large fields.
-
extractors and rank extractors for polynomial sources
Foundations of Computer Science, 2007Co-Authors: Zeev Dvir, Ariel Gabizon, Avi WigdersonAbstract:In this paper we construct explicit deterministic extractors from polynomial sources, namely from distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources. A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields. The first step towards extraction is a construction o/rank extractors, which are polynomial mappings that "extract" the algebraic rank from any system of low degree polynomials. More precisely, for any n polynomials, k of which are algebraically independent, a rank extractor outputs k algebraically independent polynomials of slightly higher degree. A result of Wooley allows us to relate algebraic rank and min-entropy and to show that a rank extractor is also a high quality condenser for polynomial sources over polynomially large fields. Finally, to turn this condenser into an extractor, we employ a theorem of Bombieri, giving a Character Sum estimate for polynomials defined over curves. It allows extracting all the randomness (up to a multiplicative constant) from polynomial sources over exponentially large fields.
Zeev Dvir - One of the best experts on this subject based on the ideXlab platform.
-
Extractors And Rank Extractors For Polynomial Sources
Computational Complexity, 2009Co-Authors: Zeev Dvir, Ariel Gabizon, Avi WigdersonAbstract:In this paper we construct explicit deterministic extractors from polynomial sources, which are distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources (which are degree 1 polynomials). A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields. The steps in our extractor construction, and the tools (mainly from algebraic geometry) that we use for them, are of independent interest: The first step is a construction of rank extractors, which are polynomial mappings which ‘extract’ the algebraic rank from any system of low degree polynomials. More precisely, for any n polynomials, k of which are algebraically independent, a rank extractor outputs k algebraically independent polynomials of slightly higher degree. The rank extractors we construct are applicable not only over finite fields but also over fields of Characteristic zero. The next step is relating algebraic independence to min-entropy. We use a theorem of Wooley to show that these parameters are tightly connected. This allows replacing the algebraic asSumption on the source (above) by the natural information theoretic one. It also shows that a rank extractor is already a high quality condenser for polynomial sources over polynomially large fields. Finally, to turn the condensers into extractors, we employ a theorem of Bombieri, giving a Character Sum estimate for polynomials defined over curves. It allows extracting all the randomness (up to a multiplicative constant) from polynomial sources over exponentially large prime fields.
-
FOCS - Extractors and Rank Extractors for Polynomial Sources
48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07), 2007Co-Authors: Zeev Dvir, Ariel Gabizon, Avi WigdersonAbstract:In this paper we construct explicit deterministic extractors from polynomial sources, namely from distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources. A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields. The first step towards extraction is a construction o/rank extractors, which are polynomial mappings that "extract" the algebraic rank from any system of low degree polynomials. More precisely, for any n polynomials, k of which are algebraically independent, a rank extractor outputs k algebraically independent polynomials of slightly higher degree. A result of Wooley allows us to relate algebraic rank and min-entropy and to show that a rank extractor is also a high quality condenser for polynomial sources over polynomially large fields. Finally, to turn this condenser into an extractor, we employ a theorem of Bombieri, giving a Character Sum estimate for polynomials defined over curves. It allows extracting all the randomness (up to a multiplicative constant) from polynomial sources over exponentially large fields.
-
extractors and rank extractors for polynomial sources
Foundations of Computer Science, 2007Co-Authors: Zeev Dvir, Ariel Gabizon, Avi WigdersonAbstract:In this paper we construct explicit deterministic extractors from polynomial sources, namely from distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources. A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields. The first step towards extraction is a construction o/rank extractors, which are polynomial mappings that "extract" the algebraic rank from any system of low degree polynomials. More precisely, for any n polynomials, k of which are algebraically independent, a rank extractor outputs k algebraically independent polynomials of slightly higher degree. A result of Wooley allows us to relate algebraic rank and min-entropy and to show that a rank extractor is also a high quality condenser for polynomial sources over polynomially large fields. Finally, to turn this condenser into an extractor, we employ a theorem of Bombieri, giving a Character Sum estimate for polynomials defined over curves. It allows extracting all the randomness (up to a multiplicative constant) from polynomial sources over exponentially large fields.
Lillian B. Pierce - One of the best experts on this subject based on the ideXlab platform.
-
Burgess bounds for short Character Sums evaluated at forms II: the mixed case
arXiv: Number Theory, 2020Co-Authors: Lillian B. PierceAbstract:This work proves a Burgess bound for short mixed Character Sums in $n$ dimensions. The non-principal multiplicative Character of prime conductor $q$ may be evaluated at any "admissible" form, and the additive Character may be evaluated at any real-valued polynomial. The resulting upper bound for the mixed Character Sum is nontrivial when the length of the Sum is at least $q^{\beta}$ with $\beta> 1/2 - 1/(2(n+1))$ in each coordinate. This work capitalizes on the recent stratification of multiplicative Character Sums due to Xu, and the resolution of the Vinogradov Mean Value Theorem in arbitrary dimensions.
-
Burgess bounds for multi-dimensional short mixed Character Sums
Journal of Number Theory, 2016Co-Authors: Lillian B. PierceAbstract:Abstract This paper proves Burgess bounds for short mixed Character Sums in multi-dimensional settings. The mixed Character Sums we consider involve both an exponential evaluated at a real-valued multivariate polynomial f, and a product of multiplicative Dirichlet Characters. We combine a multi-dimensional Burgess method with recent results on multi-dimensional Vinogradov Mean Value Theorems for translation–dilation invariant systems in order to prove Character Sum bounds in k ≥ 1 dimensions that recapture the Burgess bound in dimension 1. Moreover, we show that by embedding any given polynomial f into an advantageously chosen translation–dilation invariant system constructed in terms of f, we may in many cases significantly improve the bound for the associated Character Sum, due to a novel phenomenon that occurs only in dimensions k ≥ 2 .
-
Burgess bounds for multi-dimensional short mixed Character Sums
arXiv: Number Theory, 2014Co-Authors: Lillian B. PierceAbstract:This paper proves Burgess bounds for short mixed Character Sums in multi-dimensional settings. The mixed Character Sums we consider involve both an exponential evaluated at a real-valued multivariate polynomial, and a product of multiplicative Dirichlet Characters. We combine a multi-dimensional Burgess method with recent results on multi-dimensional Vinogradov Mean Value Theorems for translation-dilation invariant systems in order to prove Character Sum bounds in any dimension that recapture the Burgess bound in dimension 1. Moreover, we show that by embedding the given polynomial into an advantageously chosen translation-dilation invariant system constructed in terms of that polynomial, we may in many cases significantly improve the bound for the associated Character Sum, due to a novel phenomenon that occurs only in dimensions two and higher.
Ariel Gabizon - One of the best experts on this subject based on the ideXlab platform.
-
Extractors And Rank Extractors For Polynomial Sources
Computational Complexity, 2009Co-Authors: Zeev Dvir, Ariel Gabizon, Avi WigdersonAbstract:In this paper we construct explicit deterministic extractors from polynomial sources, which are distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources (which are degree 1 polynomials). A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields. The steps in our extractor construction, and the tools (mainly from algebraic geometry) that we use for them, are of independent interest: The first step is a construction of rank extractors, which are polynomial mappings which ‘extract’ the algebraic rank from any system of low degree polynomials. More precisely, for any n polynomials, k of which are algebraically independent, a rank extractor outputs k algebraically independent polynomials of slightly higher degree. The rank extractors we construct are applicable not only over finite fields but also over fields of Characteristic zero. The next step is relating algebraic independence to min-entropy. We use a theorem of Wooley to show that these parameters are tightly connected. This allows replacing the algebraic asSumption on the source (above) by the natural information theoretic one. It also shows that a rank extractor is already a high quality condenser for polynomial sources over polynomially large fields. Finally, to turn the condensers into extractors, we employ a theorem of Bombieri, giving a Character Sum estimate for polynomials defined over curves. It allows extracting all the randomness (up to a multiplicative constant) from polynomial sources over exponentially large prime fields.
-
FOCS - Extractors and Rank Extractors for Polynomial Sources
48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07), 2007Co-Authors: Zeev Dvir, Ariel Gabizon, Avi WigdersonAbstract:In this paper we construct explicit deterministic extractors from polynomial sources, namely from distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources. A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields. The first step towards extraction is a construction o/rank extractors, which are polynomial mappings that "extract" the algebraic rank from any system of low degree polynomials. More precisely, for any n polynomials, k of which are algebraically independent, a rank extractor outputs k algebraically independent polynomials of slightly higher degree. A result of Wooley allows us to relate algebraic rank and min-entropy and to show that a rank extractor is also a high quality condenser for polynomial sources over polynomially large fields. Finally, to turn this condenser into an extractor, we employ a theorem of Bombieri, giving a Character Sum estimate for polynomials defined over curves. It allows extracting all the randomness (up to a multiplicative constant) from polynomial sources over exponentially large fields.
-
extractors and rank extractors for polynomial sources
Foundations of Computer Science, 2007Co-Authors: Zeev Dvir, Ariel Gabizon, Avi WigdersonAbstract:In this paper we construct explicit deterministic extractors from polynomial sources, namely from distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources. A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields. The first step towards extraction is a construction o/rank extractors, which are polynomial mappings that "extract" the algebraic rank from any system of low degree polynomials. More precisely, for any n polynomials, k of which are algebraically independent, a rank extractor outputs k algebraically independent polynomials of slightly higher degree. A result of Wooley allows us to relate algebraic rank and min-entropy and to show that a rank extractor is also a high quality condenser for polynomial sources over polynomially large fields. Finally, to turn this condenser into an extractor, we employ a theorem of Bombieri, giving a Character Sum estimate for polynomials defined over curves. It allows extracting all the randomness (up to a multiplicative constant) from polynomial sources over exponentially large fields.