On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.
-
Upload
kendall-batson -
Category
Documents
-
view
217 -
download
3
Transcript of On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.
![Page 1: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/1.jpg)
On the degree of symmetric functions on
the Boolean cubeJoint work with Amir
Shpilka
![Page 2: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/2.jpg)
The basic question of complexity
: 0,1 0,1n
f
![Page 3: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/3.jpg)
The basic question of complexity
: 0,1 0,1n
f
How complex is it (how hard it is to compute f?)
![Page 4: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/4.jpg)
The basic question of complexity
: 0,1 0,1n
f
How complex is it (how hard it is to compute f?)
That depends on the computational model at hand. e.g. Turing machines, Circuits, Decision trees, etc…
![Page 5: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/5.jpg)
Polynomials as computers
: 0,1 0,1n
f
How complex is it (how hard it is to compute f?)
That depends on the computational model at hand. e.g. Turing machines, Circuits, Decision trees, etc…
Our model of computation – Polynomials.
![Page 6: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/6.jpg)
Polynomials as computers
: 0,1 0,1n
f
1, , np x x
0,1n
x p x f x
Our model of computation – Polynomials.
![Page 7: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/7.jpg)
Polynomials as computers
: 0,1 0,1n
f
1, , np x x
0,1n
x p x f x
Our model of computation – Polynomials.
Complexity of is deg degf f p
![Page 8: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/8.jpg)
Tight lower boundNisan and Szegedy (94) proved
assuming f depend on all n variables.
2deg log log logf n O n
![Page 9: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/9.jpg)
Tight lower boundNisan and Szegedy (94) proved
assuming f depend on all n variables.
Can we get stronger lower bounds on more restricted natural classes of functions?
2deg log log logf n O n
![Page 10: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/10.jpg)
Symmetric Boolean functions
1 1, , , ,n n nS f x x f x x
![Page 11: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/11.jpg)
Symmetric Boolean functions
Von zur Gathen and Roche (97) proved
assuming f is non-constant.
1 1, , , ,n n nS f x x f x x
0.525deg f n O n
![Page 12: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/12.jpg)
Symmetric Boolean functions
1 1, , n nf x x F x x
![Page 13: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/13.jpg)
Symmetric Boolean functions
1 1, , n nf x x F x x
: 0,1 0,1 : symmetric : 0,1, , 0,1n
f f F n
![Page 14: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/14.jpg)
Symmetric Boolean functions
: 0,1, , 0,1,2, ,f n c
0 1 2 3 4 5 6 . . . n012
.
.
.
c
![Page 15: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/15.jpg)
Symmetric Boolean functions
: 0,1, , 0,1,2, ,f n c
0 1 2 3 4 5 6 . . . n012
.
.
.
c
![Page 16: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/16.jpg)
Symmetric Boolean functions
: 0,1, , 0,1,2, ,f n c
deg fWhat can be said about ?
0 1 2 3 4 5 6 . . . n012
.
.
.
c
![Page 17: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/17.jpg)
Symmetric functions
: 0,1, , 0,1,2, ,f n c
deg fWhat can be said about ?
For c=1 we got
For c=n the function has degree 1.
deg f n o n
f k k
![Page 18: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/18.jpg)
Symmetric functions
: 0,1, , 0,1,2, ,f n c
deg fWhat can be said about ?
For c=1 we got
For c=n the function has degree 1.
How does the degree behaves?
deg f n o n
f k k
![Page 19: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/19.jpg)
Symmetric functions
Von zur Gathen and Roche noted that
1deg
1
nf
c
![Page 20: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/20.jpg)
Symmetric functions
Von zur Gathen and Roche noted that
In particular, even for this observation doesn’t exclude the existence of a parabola interpolating on some function.
1deg
1
nf
c
/ 2c n
![Page 21: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/21.jpg)
Relative degree
: 0,1, , 0,1,2, ,f n c
Define
1min deg : as abovecD n f fn
![Page 22: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/22.jpg)
Relative degree
: 0,1, , 0,1,2, ,f n c
Define
is monotone decreasing in c.
1min deg : as abovecD n f fn
cD n
![Page 23: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/23.jpg)
Relative degree
: 0,1, , 0,1,2, ,f n c
Define
is monotone decreasing in c.
has a crazy behavior in n.
1min deg : as abovecD n f fn
cD n
cD n
![Page 24: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/24.jpg)
Relative degree
: 0,1, , 0,1,2, ,f n c
Define
is monotone decreasing in c.
has a crazy behavior in n.
1min deg : as abovecD n f fn
cD n
cD n
1
1cD nc
![Page 25: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/25.jpg)
6 stages of first-time research
Stage 1
![Page 26: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/26.jpg)
6 stages of first-time research
Stage 2
![Page 27: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/27.jpg)
6 stages of first-time research
Stage 3
![Page 28: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/28.jpg)
6 stages of first-time research
Stage 4
![Page 29: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/29.jpg)
6 stages of first-time research
Stage 5
![Page 30: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/30.jpg)
6 stages of first-time research
Stage 6
![Page 31: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/31.jpg)
6 stages of first-time research
Stage
1…
![Page 32: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/32.jpg)
Our main result
1
91
22nD n o
Main theorem
This proves a threshold behavior at c=n.
![Page 33: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/33.jpg)
Main theorem
This proves a threshold behavior at c=n.
Yet another theorem
Our main result
1
91
22nD n o
: 0,1,..., , 1f n C C O
2deg
3f n o n
![Page 34: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/34.jpg)
Proof strategy – reducing c
2n o n p n Lemma 1. For any n there exist a prime p such that and
1 4
11
2nD n D p o
![Page 35: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/35.jpg)
Proof strategy – reducing c
2n o n p n Lemma 1. For any n there exist a prime p such that and
Together with the trivial bound , we already get a threshold behavior
1 4
11
2nD n D p o
4 1/ 5D p
1
11
10nD n o
![Page 36: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/36.jpg)
Proof strategy – reducing nn mLemma 2. For every c,m,n such that , it
holds that
c cD n D m
Dream
version
![Page 37: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/37.jpg)
Proof strategy – reducing nn mLemma 2. For every c,m,n such that , it
holds that
1c cD n D m o
Dream
version
![Page 38: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/38.jpg)
Proof strategy – reducing nn mLemma 2. For every c,m,n such that , it
holds that
11c c
mD n D m o
m
Dream
version
![Page 39: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/39.jpg)
Proof strategy – reducing n2mn cLemma 2. For every c,m,n such that , it
holds that
11c c
mD n D m o
m
![Page 40: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/40.jpg)
Proof of the main theorem
A computer search found that .By Lemma 2
By Lemma 1
4 21 6 / 7D
4
21 6 91 1
21 1 7 11D n o o
1 4
1 1 9 91 1 1
2 2 11 22nD n D p o o o
![Page 41: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/41.jpg)
Periodicity and degree
Low degree Strong periodical structure
Dream
version
![Page 42: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/42.jpg)
Periodicity and degree
Low degree Strong periodical structure
Strong periodical structure High degree
Dream
version
![Page 43: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/43.jpg)
Periodicity and degree
Low degree Strong periodical structure
Strong periodical structure High degree
Hence no function has “to low” degree.
Dream
version
![Page 44: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/44.jpg)
Periodicity and degree
Low degree Strong periodical structure
Strong periodical structure High degree
Not the same sense of periodical structure…
![Page 45: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/45.jpg)
Low degree implies strong periodical structure
pf p j f j
Lemma 3. Let with . Let be a prime number. Then for all such that it holds that
: 0,1, , 0,1, ,f n c
deg f d d p n 0 j d
0 1 2 3 . . . d . . . p
01
.
.
c
n
p j n
![Page 46: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/46.jpg)
Low degree implies strong periodical structure
pf p j f j
Lemma 3. Let with . Let be a prime number. Then for all such that it holds that
: 0,1, , 0,1, ,f n c
deg f d d p n 0 j d
0 1 2 3 . . . d . . . p q
01
.
.
c
n
p j n
![Page 47: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/47.jpg)
Low degree implies strong periodical structure
pf p j f j
Lemma 3. Let with . Let be a prime number. Then for all such that it holds that
: 0,1, , 0,1, ,f n c
deg f d d p n 0 j d
0 1 2 3 . . . d . . . p q r
01
.
.
c
n
p j n
![Page 48: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/48.jpg)
Strong periodical structure implies high degree
0 :TP f k n T f k f k T
Definition. Let and define
: 0,1, , 0,1, ,f n c 1T
![Page 49: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/49.jpg)
Strong periodical structure implies high degree
0 :TP f k n T f k f k T
Definition. Let and define
: 0,1, , 0,1, ,f n c
, 10 T
![Page 50: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/50.jpg)
Lemma 4. Let . Then for all
If then
If then or
Strong periodical structure implies high degree
: 0,1, , 0,1, ,f n c
0 :TP f k n T f k f k T
Definition. Let and define
: 0,1, , 0,1, ,f n c
0, 1T
0
0
1T
deg Tf P f
deg Tf P f deg 1f
![Page 51: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/51.jpg)
Proof of Lemma 1
2n o n p n Lemma 1. For any n there exist a prime p such that and
1 4
11
2nD n D p o
![Page 52: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/52.jpg)
Proof of Lemma 1
0 1 2 . . . n012
.
.
.
n-1
f
![Page 53: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/53.jpg)
Proof of Lemma 1
0 1 2 . . . p . . . 2p n012
.
.
.
n-1
o(n)
f
![Page 54: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/54.jpg)
Proof of Lemma 1
0 1 2 . . . p . . . 2p n012
.
.
.
n-1
o(n)
f
![Page 55: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/55.jpg)
Proof of Lemma 1
0 1 2 . . . p . . . 2p n012
.
.
.
n-1
o(n)
We might as well assume that
non-constantf
f
![Page 56: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/56.jpg)
Proof of Lemma 1
0 1 2 . . . p . . . 2p n012
.
.
.
n-1
o(n)
We might as well assume that
non-constant deg f pf
f
![Page 57: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/57.jpg)
Proof of Lemma 1
Define
2 0,1,...,f p k f k
g k k pp
![Page 58: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/58.jpg)
Proof of Lemma 1
Define
From Lemma 3
2 0,1,...,f p k f k
g k k pp
0,1,...,pf p k f k k p
![Page 59: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/59.jpg)
Proof of Lemma 1
Define
From Lemma 3
and also
2 0,1,...,f p k f k
g k k pp
0,1,...,pf p k f k k p
, 0,1, 2,..., 1 2f p k f k n n p o p
![Page 60: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/60.jpg)
Proof of Lemma 1
0,1,...,pf p k f k k p
From Lemma 3
Hence : 0,1, , 0,1, 2,3, 4g p
![Page 61: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/61.jpg)
Proof of Lemma 1
Case 1: g is a non-constant
and we are done.
2f p k f k
g kp
1 4 4deg deg deg2n
nn D n f f g p D p o n D p
![Page 62: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/62.jpg)
Proof of Lemma 1
Case 2: g is a constant G
Hence , by Lemma 4
2f p k f k
g kp
21 deg deg
2G p
n p
nn D n f f P f p o n
2 0,1,...,f p k f k G p k p
![Page 63: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/63.jpg)
Proof of Lemma 1
Case 2: g is a constant G
Hence , by Lemma 4
or is linear.
2f p k f k
g kp
2 0,1,...,f p k f k G p k p
f
21 deg deg
2G p
n p
nn D n f f P f p o n
![Page 64: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/64.jpg)
Proof of Lemma 1
Case 2: If happens to be linear, apply the proof so far on .Since we are done unless it also happens that is linear.
But this means f itself must be linear. Since f is not constant it means f assumes n+1 distinct values – a contradiction.
2f p k f k
g kp
Rf k f n k
f
deg deg Rf fRf
![Page 65: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/65.jpg)
Open Questions Main question - Better understand .
Improve the lower bounds to non-linear, if possible.
cD n
![Page 66: On the degree of symmetric functions on the Boolean cube Joint work with Amir Shpilka.](https://reader038.fdocuments.in/reader038/viewer/2022103112/551b14f55503465e7d8b6243/html5/thumbnails/66.jpg)
Thank you!