We introduce a new family of propositional proof systems, denoted $\langle EF, R \rangle$, for an arbitrary TFNP search problem $R$. Informally, a refutation of a CNF formula $F$ in $\langle EF, R \rangle$ is given by a polynomial-time mapping reduction from the false-clause search problem ${\mathrm{Search}}_F$ to $R$, combined ... more >>>

We design a deterministic algorithm that, given blackbox access to the product $f=\prod_{i=1}^{\ell}{h_i}$ of $\ell$ irreducible $s$-sparse $n$-variate polynomials of bounded individual degree $d$, over fields of characteristic zero, and more generally over fields of sufficiently large positive characteristic, recovers the $h_i$'s and their multiplicities in time $\mathrm{poly}(n,(s\ell d)^d)$. ... more >>>

The Hamiltonian Cycle polynomial, denoted as $HC_n$, is defined to be the sum of the weighted Hamiltonian Cycles in an $n$-vertex complete digraph, with vertices labeled $1$ to $n$ and edges weighted by formal variables $x_{i,j}$. The Permanent and $HC$, defined as the family $\{HC_n | \ n \geq 1\}$, ... more >>>