# If a sparse language is NP complete, then are all languages in NP in P/poly?

If a Sparse Language is NP complete, then are all languages in NP in P/poly? I know that sparse languages are in P/poly, but does a polynomial time reduction give an addition to the circuit that is super polynomial? A sparse language is such that $$\forall n \in \mathbb{N}, |L \cap \{ 0,1 \} ^{n} | \leq p(n)$$ for some polynomial $$p(n)$$.

Yes, see Mahaney's theorem. It states that if a sparse language is $$\mathbf{NP}-$$complete, then $$\mathbf{P=NP}$$, and since $$\mathbf{P\subseteq P_{poly}}$$, then $$\mathbf{NP\subseteq P_{poly}}$$