TY - JOUR
T1 - Quasi-random multilinear polynomials
AU - Kalai, Gil
AU - Schulman, Leonard J.
N1 - Publisher Copyright:
© 2019, The Hebrew University of Jerusalem.
PY - 2019/3/1
Y1 - 2019/3/1
N2 - We consider multilinear Littlewood polynomials, polynomials in n variables in which a specified set of monomials U have ±1 coefficients, and all other coefficients are 0. We provide upper and lower bounds (which are close for U of degree below log n) on the minimum, over polynomials h consistent with U, of the maximum of |h| over ±1 assignments to the variables. (This is a variant of a question posed by Erdős regarding the maximum on the unit disk of univariate polynomials of given degree with unit coefficients.) We outline connections to the theory of quasi-random graphs and hypergraphs, and to statistical mechanics models. Our methods rely on the analysis of the Gale–Berlekamp game; on the constructive side of the generic chaining method; on a Khintchine-type inequality for polynomials of degree greater than 1; and on Bernstein’s approximation theory inequality.
AB - We consider multilinear Littlewood polynomials, polynomials in n variables in which a specified set of monomials U have ±1 coefficients, and all other coefficients are 0. We provide upper and lower bounds (which are close for U of degree below log n) on the minimum, over polynomials h consistent with U, of the maximum of |h| over ±1 assignments to the variables. (This is a variant of a question posed by Erdős regarding the maximum on the unit disk of univariate polynomials of given degree with unit coefficients.) We outline connections to the theory of quasi-random graphs and hypergraphs, and to statistical mechanics models. Our methods rely on the analysis of the Gale–Berlekamp game; on the constructive side of the generic chaining method; on a Khintchine-type inequality for polynomials of degree greater than 1; and on Bernstein’s approximation theory inequality.
UR - http://www.scopus.com/inward/record.url?scp=85059457577&partnerID=8YFLogxK
U2 - 10.1007/s11856-018-1821-y
DO - 10.1007/s11856-018-1821-y
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:85059457577
SN - 0021-2172
VL - 230
SP - 195
EP - 211
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -