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Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size. In a recent work of Chatterjee and the authors (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.
The problem of photoemission from a quasi-1D material is studied. We identify two issues that play a key role in the detection of gapless Tomonaga-Luttinger liquid (TLL) phase. Firstly, we show how a disorder -- backward scattering as well as forward
A subset $A$ of a semigroup $S$ is called a $chain$ ($antichain$) if $xyin{x,y}$ ($xy otin{x,y}$) for any (distinct) elements $x,yin S$. A semigroup $S$ is called ($anti$)$chain$-$finite$ if $S$ contains no infinite (anti)chains. We prove that each a
The random k-SAT model is the most important and well-studied distribution over k-SAT instances. It is closely connected to statistical physics; it is used as a testbench for satisfiability algorithms, and average-case hardness over this distribution
Software engineering (SE) undergoes an ontological crisis and it lacks of a theory. Why? Among other reasons, because always it succumbed to the pragmatism demanded by the commercial and political interests and abandoned any intention to become a sci
Automatic translation from natural language descriptions into programs is a longstanding challenging problem. In this work, we consider a simple yet important sub-problem: translation from textual descriptions to If-Then programs. We devise a novel n