By the breakthrough work of Håstad, several constraint satisfaction
problems are now known to have the following approximation resistance
property: satisfying more clauses than what picking a random
assignment would achieve is NP-hard. This is the case for example for
Max E3-Sat, Max E3-Lin and Max E4-Set Splitting. A notable ... more >>>

Elementary symmetric polynomials $S_n^k$ are used as a
benchmark for the bounded-depth arithmetic circuit model of computation.
In this work we prove that $S_n^k$ modulo composite numbers $m=p_1p_2$
can be computed with much fewer multiplications than over any field, if
the coefficients of monomials $x_{i_1}x_{i_2}\cdots x_{i_k}$ ... more >>>

The use of Nepomnjascij's Theorem in the proofs of independence results
for bounded arithmetic theories is investigated. Using this result and similar ideas, the following statements are proven: (1) At least one of S_1 or TLS does not prove the Matiyasevich-Davis-Robinson-Putnam Theorem and (2) TLS does not prove Sigma^b_{1,1}=Pi^b_{1,1}. Here ... more >>>