Let $X=X_1\sqcup X_2\sqcup\ldots\sqcup X_k$ be a partitioned set of variables such that the variables in each part $X_i$ are noncommuting but for any $i\neq j$, the variables $x \in X_i$ commute with the variables $x' \in X_j$. Given as input a square matrix $T$ whose entries are linear forms over ... more >>>

Propositional proof complexity deals with the lengths of polynomial-time verifiable proofs for Boolean tautologies. An abundance of proof systems is known, including algebraic and semialgebraic systems, which work with polynomial equations and inequalities, respectively. The most basic algebraic proof system is based on Hilbert's Nullstellensatz (Beame et al., 1996). Tropical ... more >>>

Lifting theorems are theorems that bound the communication complexity
of a composed function $f\circ g^{n}$ in terms of the query complexity
of $f$ and the communication complexity of $g$. Such theorems
constitute a powerful generalization of direct-sum theorems for $g$,
and have seen numerous applications in recent years.

We prove ... more >>>