TY - JOUR
T1 - Essential covers of the cube by hyperplanes
AU - Linial, Nathan
AU - Radhakrishnan, Jaikumar
PY - 2005/2
Y1 - 2005/2
N2 - A set L of linear polynomials in variables X1, X2, ..., Xn with real coefficients is said to be an essential cover of the cube {0, 1}n if (E1) for each ν ∈ {0, 1}n, there is a p ∈ L such that p (ν) = 0; (E2) no proper subset of L satisfies (E1), that is, for every p ∈ L, there is a ν ∈ {0, 1}n such that p alone takes the value 0 on v; (E3) every variable appears (in some monomial with non-zero coefficient) in some polynomial of L. Let e(n) be the size of the smallest essential cover of {0, 1}n. In the present note we show that 1/2 (√4n+1 + 1) le; e(n) ≤ ⌈n/2⌉ + 1.
AB - A set L of linear polynomials in variables X1, X2, ..., Xn with real coefficients is said to be an essential cover of the cube {0, 1}n if (E1) for each ν ∈ {0, 1}n, there is a p ∈ L such that p (ν) = 0; (E2) no proper subset of L satisfies (E1), that is, for every p ∈ L, there is a ν ∈ {0, 1}n such that p alone takes the value 0 on v; (E3) every variable appears (in some monomial with non-zero coefficient) in some polynomial of L. Let e(n) be the size of the smallest essential cover of {0, 1}n. In the present note we show that 1/2 (√4n+1 + 1) le; e(n) ≤ ⌈n/2⌉ + 1.
UR - http://www.scopus.com/inward/record.url?scp=14544274162&partnerID=8YFLogxK
U2 - 10.1016/j.jcta.2004.07.012
DO - 10.1016/j.jcta.2004.07.012
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AN - SCOPUS:14544274162
SN - 0097-3165
VL - 109
SP - 331
EP - 338
JO - Journal of Combinatorial Theory. Series A
JF - Journal of Combinatorial Theory. Series A
IS - 2
ER -