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+