Partial Derivatives in
Arithmetic
Complexity
and Beyond
Xi Chen
Columbia University
USA
xichen @cs. columbia.edu
Neeraj Kayal
Microsoft Research
India
neera ka
<§ microsoft, com
Avi
Wigderson
Institute for Advanced
Study
USA
avi@ias. edu
n«w
the
essence
of
knowledge
Boston
-
Delft
Contents
1
Introduction
1
1.1
Motivation
1
1.2
Arithmetic Circuits
4
1.3
Formal Derivatives and Their Properties
9
Part I: Structure
2
Symmetries of
13
a
Polynomial
17
2.1
Proof of Lemma 2.1
21
2.2
Proof of Lemma 2.4
25
3
Algebraic Independence
27
4
Polynomials
33
5
Bezout's Theorem
with
High Arithmetic Complexity
5.1
A
5.2
Proof of Theorem 5.2
5.3
Proof of Lemma 5.3
5.4
Bezout's Theorem
37
Property of Isolated Roots
over
38
39
42
Finite Fields and
ix
Rings
43
5.5
A Brief Introduction to the
5.6
6
44
Sketch of Proof of Theorem 5.6
46
Algebraic Extractors
Jacobian
6.1
p-adic Numbers,
Norm, and Metric
and the
Conjecture
The Jacobian
7
The "Joints
8
The
49
50
Conjecture
Conjecture" Resolved
53
Stepanov Method
55
8.1
Weil's Theorem on Rational Points
8.2
A
8.3
Formal Proof of Weil's Theorem
66
8.4
The Heath-Brown and Mit'kin Estimates
70
on
Curves
High-level Description of the Stepanov Method
56
62
Part II: Lower Bounds
73
9
75
General Arithmetic Circuits
10 Sums of Powers of Linear Forms
10.1
Arithmetic Circuits with Addition
and
10.2
11
85
Powering Gates
Depth-2 Symmetric Arithmetic Circuits
Depth-3 Arithmetic Circuits
88
88
91
12 Arithmetic Formulae
95
12.1 Cover Sets and Measure Functions
97
12.2 A
98
Constant-depth
Lower Bound
13
Projections
Part III:
14
of Determinant to Permanent
Algorithms
109
Identity Testing
111
14.1 POLYDEP and its Connection to
14.2 Basic
14.3 The
Identity Testing
Properties of POLYDEP
Algorithm
15 Absolute
103
112
113
114
Irreducibility Testing
119
15.1 Notation
120
15.2
122
16
Consequences
Polynomial Equivalence Testing
16.1
Algorithm
16.2
Equivalence
to a Sum of Powers
16.3
Equivalence
to
for
Minimizing the Number of Variables
Acknowledgments
References
an
Elementary Symmetric Polynomial
125
128
130
132
139