goal:
	algebraic circuit complexity
	determinant vs permanent
recall
	Q. P vs NP
		Turing machines		relativization barrier!
		circuits		natural proofs barrier!
		???
		algebra?		[[perhaps no barriers?]]
polynomials
	def
	Q. which are easy to compute?
	def
		algebraic circuits
	rmk
		non-uniform model
		practical
			[[most(ish) upper bounds for polynomials are (or start with) an alg ckts]]
		"restriction" of boolean ckts
			treat numbers as numbers
			[[*might* be easier than boolean ckt lbs]]
		impractical
			allow *arbitrary* constants
			x^{2^n} is "easy" here
				2^{2^n} is large
		computing as formal polynomial
			x^2-x\not\equiv 0, even over \F_2 where they are equiv functions
				[[restricted model of boolean function computation]]
	ex
		polynomial multiplication
			def
			best algorithms are algebraic
				related to integer multiplication
			n^2 size obvious
			O(n\log n\log\log n) over any field via FFT
		matrix multiplication
			def
			best algorithms are algebraic
			n^3 size obvious
			n^{2.81} via strassen
				find better for n=2, use divide and conqueror
			n^{2.373..} best known
		determinant
			def
			best algorithms are algebraic
			poly(n!)-size naive algorithm
			gaussian elimination
				poly(n)-size algebraic circuit over +,\times,\div
				thm[Strassen]: elimate divisions (with poly(degree) blowup)
				=> poly(n)-size algebraic circuit over +,\times
			manipulations of characteristic polynomial
				poly(n)-size algebraic circuit over +,\times
		permanent
			def
			best algorithms are algebraic
			poly(n!)-size naive algorithm
			poly(n)2^n-size Ryser formula
				[[best known]]
			thm[Toda]:
			Q. perm in P?
				P/poly?
				poly-size algebraic circuit?
					=>_GRH NP\subseteq P/poly
structural results
	def
		VP
		size
		degree
	ex
		det_n
			on n^2 variables
		not x^{2^n}
	rmk:
		poly-degree
			most interesting polynomials are polynomial degree
			x^{2^n} cannot be evaluated over the integers
			many results have poly(deg) dependence
	Q. what polynomials are in VP?
	def
		VNP
	rmk:
		more like #P
	Q. VP vs VNP
	def:
		projection reduction
		VNP completeness
	Thm[Valiant]: over fields of char!=2, perm is VNP complete
	rmk:
		in char 2, perm=det
		VP=VNP iff perm\in VP
	Thm[VSBR]: det is quasi VP-complete
		depth reduction for formulas
		depth reduction for ckts
		formulas \le det
	Q. det vs perm
next time
	lbs for restricted models