On the Sum of Square Roots of Polynomials and
... Pn 1.1√(Lower bounding a non-zero ‘signed’ sum of square roots of integers). Given a sum S = i=1 δi · ai , where δi ∈ {+1, −1} and ai ’s are positive integers upper bounded by N , find a tight lower bound on |S| in terms of n and N when S 6= 0. Is it true that for a non-zero S, |S| ≥ 1/2poly(n,log N ...
... Pn 1.1√(Lower bounding a non-zero ‘signed’ sum of square roots of integers). Given a sum S = i=1 δi · ai , where δi ∈ {+1, −1} and ai ’s are positive integers upper bounded by N , find a tight lower bound on |S| in terms of n and N when S 6= 0. Is it true that for a non-zero S, |S| ≥ 1/2poly(n,log N ...
Solutions
... 2. By substituting x = 2 in the quotient ring Z[x]/(x2 − 7, x − 2) we obtain that it is the field Z/(3). So x − 2, and similarily x + 2 is a maximal ideal. These two maximal ideals are not associates. Because, assume that we have a + bx such that 2 + x = (2 − x)(a + bx) = 2a − 7b + x(2b − a). This i ...
... 2. By substituting x = 2 in the quotient ring Z[x]/(x2 − 7, x − 2) we obtain that it is the field Z/(3). So x − 2, and similarily x + 2 is a maximal ideal. These two maximal ideals are not associates. Because, assume that we have a + bx such that 2 + x = (2 − x)(a + bx) = 2a − 7b + x(2b − a). This i ...
