Essential covers of the cube by hyperplanes

A set L of linear polynomials in variables X 1 , X 2 , … , X n with real coefficients is said to be an essential cover of the cube { 0 , 1 } n if(E1) ch v ∈ { 0 , 1 } n , there is a p ∈ L such that p ( v ) = 0 ; per subset of L satisfies (E1), that is, for every p ∈ L , there is a v ∈ { 0 , 1 } n such that p alone takes the value 0 on v; variable appears (in some monomial with non-zero coefficient) in some polynomial of L. ( n ) be the size of the smallest essential cover of { 0 , 1 } n . In the present note we show that 1 2 ( 4 n + 1 + 1 ) ⩽ e ( n ) ⩽ n 2 + 1 .