Quadratic-Solvability

1
Quadratic-Solvability

• All you need is gap

2
Def Quadratic-Solvability

• An instance of Quadratic-Solvability ( QS? )
  is a system of quadratic-polynomials P1, , Pn
  over a field ? and variables y1, , ym (where m
  ? n)
• Problemto decide whether there exists a common
  root to all P1, , Pn
• Thm GJ QS? is NP-hard (for any field ?)

3
Def gap-QS D, ?, ?

• An instance a system of quadratic polynomials
  P1, , Pn over a field ?, and variables y1, ,
  ym,where each Pi depends on at most D variables
• Problemdistinguish between the following two
  cases
• yes there exists a root common to all P1, , Pn
• no any assignment to y1, , yn zeros lt ?
  fraction of P1, , Pn

4
Reducing QS? to gap-QS n, ?, 2?-1

• ThmHPS gap-QS n, ?, 2?-1 is NP-hard As
  long as ? ? nc for some cgt0
• Proof Given an instance P1, , Pn of QS?
• set H so that H ?½
• and d so that Hd n
• associate each point x ? Hd with a distinct
  quadratic-polynomial Px ? P1, , Pn
• For any assignment A, denote by ƒA Hd ? ? the
  function describing each Pis value on A Namely,
  ƒA(x), on every x ? Hd, equals PxA

5
Extension

• Consider now LDEƒA namely, the
  low-degree-extension of ƒA to ? ?d
• As seen in one of the early home assignments,
  LDEƒAs value on any x ? ? is a linear
  combination of As value on Hd LDEƒA(x) ?y
  ? Hd cxy A(y)
• Let now Px, for every x ? ?, be Px ?y ? Hd
  cxy Py that is, a quadratic-polynomial being
  the corresponding linear combination of P1, , Pn

6
The Resulting QS Instance

• Pxx ?? is a set of quadratic-polynomials, over
  the same variables y1, , ymwhich for any
  assignment A satisfies the following
• If A zeros all Pi, then LDE ƒA is identically
  zero, and therefore A zeros Px for every x ??
• In case LDE ƒA is not identically zero, like a
  good LDF it is almost nowhere zero, hence A zeros
  at most a very small (lt r ?½)fraction of Pxx
  ??

7
Our Goal

• ThmRS,DFKRS QS O(1), ?, 2/? is NP-hardas
  long as log? ? log?n for any constant ? lt 1
  that is, the difference between Cooks
  characterization of NP and PCP characterizations
  of NP (with constant number of variables each
  local-test depends on) boils down to the number
  of variables each polynomial accesses