admin
	ps6 due
	projects:
		reports due 05-02
		presentations next week
today
	homogeneous polynomials and ckts
recall
	algebraic circuits compute polynomials formally
		x^2-x \ne 0 over \F_2[x]
	VP vs VNP = det vs perm
	fact: exist hard polynomials
	Pf-sketch
		over finite fields
	rmk:
		w/ 0,1 coefficients 
		over algebraic closure
	goal
		lower bounds for explicit polynomials
			for restricted classes
homogeneous polynomial
	def
		homogenous polynomial
		homogeneous circuit
	Q. is homogenization a restriction?
	prop: size s ckt computing f deg \le d
		=> size poly(s,d) ckt computing homogenous components of f
	pf
		by induction on gates
	rmk:
		never need to compute about \deg f
		only works for d\le poly
		increases the depth
			unbounded fan-in => fan-in-2 => unbounded fan-in
	Q. is homogenization a restriction in low-depth?
	prop: esym_d=
		has O(n^2)-size depth-3 non-homogeneous formula
	pf
		by interpolation
	rmk:
		surprising:
			esym =~ MAJ
			\parity\notin \AC => \MAJ\notin \AC
						\notin AC[p] any constant p
			[[constant-depth algebraic ckts seem to have more power than boolean analogues]]
	prop: esym_d
		require \Omega(n/d)^d size as homogeneous depth-3 formula
		[[won't show]]
	prop: det_n requires 2^{\Omega(n)}-size homogeneous depth-3 formula
		[[will show]]
	def:
		formal partial derivative
	lem:
		linearity
		liebnitz rule
	def:
		iterated formal partial derivative
	lem:
		linearity
		liebnitz rule
	def:
		space of partial derivatives
		dimension of space
	lem:
		subadditive
		submultiplicative
	prop: dim of homog depth-3
	pf
	prop: dim of det
	pf
		set of partial derivatives
		dimension
	rmk:
		same proof for esym works
			need more linear algebra to lb dimension
		\mu(ckts) and \mu(hard poly) were *both* exponentially big
		lbs for homogeneous depth-4 only known since 2014
		no lbs for homogenous depth-5+