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UID:www.tcs.tifr.res.in/event/1718
DTSTAMP:20260514T053912Z
SUMMARY:Exponential Sums and Weil Bounds: A Survey of Elementary Methods an
 d Decoding Applications
DESCRIPTION:Speaker: Soham Chatterjee (TIFR)\n\nAbstract: \nThis talk is a 
 survey on techniques for estimating the number of solutions to polynomial 
 equations over a finite field $\\mathbb{F}_q$. We use exponential sums-con
 structed from additive and multiplicative character-to establish these cou
 nts. By analyzing the algebraic and orthogonality properties of Gaussian a
 nd Jacobi sums\, we build a mathematical framework to derive bounds for eq
 uations over $\\mathbb{F}_q$ and its extensions.A central part of the surv
 ey is the study of Dirichlet $L$-functions over the polynomial ring $\\mat
 hbb{F}_q[X]$. We demonstrate how these $L$-functions can be used with char
 acters on the set of monic rational functions to get bounds on the absolut
 e value of character sums on polynomials. We discuss the Weil bounds for a
 bsolutely irreducible bivariate polynomials\, focusing on two primary spec
 ial cases: \nSuperelliptic curves: $Y^d = f(X)$ analyzed via Stepanov's m
 ethod.\nArtin-Schreier curves: $Y^q - Y = f(X)$ analyzed via Bombieri's me
 thod. \nThese elementary polynomial methods are emphasized because they a
 chieve the same results as advanced algebraic geometry without requiring i
 ts full abstract machinery.\nThe final part of the talk details an algorit
 hmic application of these bounds: decoding codes constructed on evaluatin
 g polynomial composed with multiplicative character on whole field and dua
 l BCH codes from a constant fraction of errors. This problem is challengi
 ng because character evaluations (like the quadratic character $\\eta$) di
 scard most information about a polynomial\, leading to underdetermined sys
 tems where traditional decoding fails. To address this\, we introduce pse
 udopolynomials which are of high-degree polynomials whose Hasse derivativ
 es agree with low-degree polynomials when evaluated on $\\mathbb{F}_q$. By
  merging the Berlekamp–Welch like arguments for Reed-Solomon with the 
 interpolation techniques from Stepanov's method\, to decode from $\\frac1
 8$th of the minimum distance of the codes. The algorithm is from a recent 
 work of Kopparty. \n
URL:https://www.tcs.tifr.res.in/web/events/1718
DTSTART;TZID=Asia/Kolkata:20260522T110000
DTEND;TZID=Asia/Kolkata:20260522T120000
LOCATION:A-201 (STCS Seminar Room)
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