The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA...

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The real τ -conjecture & lower bounds for the permanent Bruno Grenet LIP ´ ENS de Lyon Rencontres CoA 22 novembre 2012

Transcript of The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA...

Page 1: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

The real τ-conjecture&

lower bounds for the permanent

Bruno GrenetLIP ENS de Lyon

Rencontres CoA 22 novembre 2012

Page 2: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Arithmetic Circuits

f (x , y , z) = x4 + 4 x3y + 6 x2y2 + 4 xy3 + x2z + 2 xyz

+ y2z + x2 + y4 + 2 xy + y2 + z2 + 2 z + 1

x y z 1

Complexity of a polynomialτ(f ) = size of its smallest circuit

representation

2 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 3: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Arithmetic Circuits

f (x , y , z) = (x + y)4 + (z + 1)2 + (x + y)2(z + 1)

+ y2z + x2 + y4 + 2 xy + y2 + z2 + 2 z + 1

x y z 1

Complexity of a polynomialτ(f ) = size of its smallest circuit

representation

2 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 4: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Arithmetic Circuits

f (x , y , z) = (x + y)4 + (z + 1)2 + (x + y)2(z + 1)

+ y2z + x2 + y4 + 2 xy + y2 + z2 + 2 z + 1

x y z 1

Complexity of a polynomialτ(f ) = size of its smallest circuit

representation

2 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 5: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Arithmetic Circuits

f (x , y , z) = (x + y)4 + (z + 1)2 + (x + y)2(z + 1)

+ y2z + x2 + y4 + 2 xy + y2 + z2 + 2 z + 1

x y

z 1

Complexity of a polynomialτ(f ) = size of its smallest circuit

representation

2 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 6: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Arithmetic Circuits

f (x , y , z) = (x + y)4 + (z + 1)2 + (x + y)2(z + 1)

+ y2z + x2 + y4 + 2 xy + y2 + z2 + 2 z + 1

x y z 1

Complexity of a polynomialτ(f ) = size of its smallest circuit

representation

2 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 7: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Arithmetic Circuits

f (x , y , z) = (x + y)4 + (z + 1)2 + (x + y)2(z + 1)

+ y2z + x2 + y4 + 2 xy + y2 + z2 + 2 z + 1

x y z 1

Complexity of a polynomialτ(f ) = size of its smallest circuit

representation

2 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 8: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Arithmetic Circuits

f (x , y , z) = (x + y)4 + (z + 1)2 + (x + y)2(z + 1)

+ y2z + x2 + y4 + 2 xy + y2 + z2 + 2 z + 1

x y z 1

Complexity of a polynomialτ(f ) = size of its smallest circuit

representation

2 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 9: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Arithmetic Circuits

f (x , y , z) = (x + y)4 + (z + 1)2 + (x + y)2(z + 1)

+ y2z + x2 + y4 + 2 xy + y2 + z2 + 2 z + 1

x y z 1

Complexity of a polynomialτ(f ) = size of its smallest circuit

representation

2 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 10: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Arithmetic Circuits

f (x , y , z) = (x + y)4 + (z + 1)2 + (x + y)2(z + 1)

+ y2z + x2 + y4 + 2 xy + y2 + z2 + 2 z + 1

x y z 1

Complexity of a polynomialτ(f ) = size of its smallest circuit

representation

2 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 11: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Arithmetic Circuits

f (x , y , z) = (x + y)4 + (z + 1)2 + (x + y)2(z + 1)

+ y2z + x2 + y4 + 2 xy + y2 + z2 + 2 z + 1

x y z 1

Complexity of a polynomialτ(f ) = size of its smallest circuit

representation

2 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 12: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Arithmetic Circuits

f (x , y , z) = (x + y)4 + (z + 1)2 + (x + y)2(z + 1)

+ y2z + x2 + y4 + 2 xy + y2 + z2 + 2 z + 1

x y z 1

Complexity of a polynomialτ(f ) = size of its smallest circuit

representation

2 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 13: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

The τ-conjectureConjecture (Shub & Smale, 1995)The number of integer roots of any f ∈ Z[X ] is ≤ poly(τ(f )).

Theorem (Burgisser, 2007)τ-conjecture

=⇒ super-polynomial lower bound for the permanent

=⇒ τ(PERn) is not polynomially bounded in n

=⇒ VP0 6= VNP0

PERn(x11, . . . , xnn) = per

x11 · · · x1n...

...xn1 · · · xnn

=∑σ∈Sn

n∏i=1

xiσ(i)

3 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 14: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

The τ-conjectureConjecture (Shub & Smale, 1995)The number of integer roots of any f ∈ Z[X ] is ≤ poly(τ(f )).

Theorem (Burgisser, 2007)τ-conjecture

=⇒ super-polynomial lower bound for the permanent

=⇒ τ(PERn) is not polynomially bounded in n

=⇒ VP0 6= VNP0

PERn(x11, . . . , xnn) = per

x11 · · · x1n...

...xn1 · · · xnn

=∑σ∈Sn

n∏i=1

xiσ(i)

3 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 15: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

The τ-conjectureConjecture (Shub & Smale, 1995)The number of integer roots of any f ∈ Z[X ] is ≤ poly(τ(f )).

Theorem (Burgisser, 2007)τ-conjecture

=⇒ super-polynomial lower bound for the permanent

=⇒ τ(PERn) is not polynomially bounded in n

=⇒ VP0 6= VNP0

PERn(x11, . . . , xnn) = per

x11 · · · x1n...

...xn1 · · · xnn

=∑σ∈Sn

n∏i=1

xiσ(i)

3 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 16: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

The τ-conjectureConjecture (Shub & Smale, 1995)The number of integer roots of any f ∈ Z[X ] is ≤ poly(τ(f )).

Theorem (Burgisser, 2007)τ-conjecture

=⇒ super-polynomial lower bound for the permanent

=⇒ τ(PERn) is not polynomially bounded in n

=⇒ VP0 6= VNP0

PERn(x11, . . . , xnn) = per

x11 · · · x1n...

...xn1 · · · xnn

=∑σ∈Sn

n∏i=1

xiσ(i)

3 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 17: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

The τ-conjectureConjecture (Shub & Smale, 1995)The number of integer roots of any f ∈ Z[X ] is ≤ poly(τ(f )).

Theorem (Burgisser, 2007)τ-conjecture

=⇒ super-polynomial lower bound for the permanent

=⇒ τ(PERn) is not polynomially bounded in n

=⇒ VP0 6= VNP0

PERn(x11, . . . , xnn) = per

x11 · · · x1n...

...xn1 · · · xnn

=∑σ∈Sn

n∏i=1

xiσ(i)

3 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 18: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

The τ-conjecture is hardTheorem (Shub & Smale, 1995)τ-conjecture =⇒ PC 6= NPC

Theorem (Cheng, 2003)Extended τ-conjecture =⇒ Merel torsion theorem, . . .

False for real roots (Shub-Smale 95, Borodin-Cook 76)Tn = n-th Chebyshev polynomialI τ(Tn) = O(log n)I n real roots

4 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 19: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

The τ-conjecture is hardTheorem (Shub & Smale, 1995)τ-conjecture =⇒ PC 6= NPC

Theorem (Cheng, 2003)Extended τ-conjecture =⇒ Merel torsion theorem, . . .

False for real roots (Shub-Smale 95, Borodin-Cook 76)Tn = n-th Chebyshev polynomialI τ(Tn) = O(log n)I n real roots

4 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 20: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

The τ-conjecture is hardTheorem (Shub & Smale, 1995)τ-conjecture =⇒ PC 6= NPC

Theorem (Cheng, 2003)Extended τ-conjecture =⇒ Merel torsion theorem, . . .

False for real roots (Shub-Smale 95, Borodin-Cook 76)Tn = n-th Chebyshev polynomialI τ(Tn) = O(log n)I n real roots

4 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 21: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Let’s make it real!Real τ-conjecture (Koiran, 2011)Let f =

∑ki=1

∏mj=1 fij where the fij ’s are t-sparse polynomials.

Then f has ≤ poly(k ,m, t) real roots.

Theorem (Koiran, 2011)Real τ-conjecture

=⇒ Super-polynomial lower bound for the permanent

I Enough to bound the number of integer roots Adelic τ-conjecture [Phillipson & Rojas, 2012]

I Case k = 1: Follows from Descartes’ rule.I Case k = 2: Open.I Toy question: Number of real roots of fg + 1?

5 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 22: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Let’s make it real!Real τ-conjecture (Koiran, 2011)Let f =

∑ki=1

∏mj=1 fij where the fij ’s are t-sparse polynomials.

Then f has ≤ poly(k ,m, t) real roots.

Theorem (Koiran, 2011)Real τ-conjecture

=⇒ Super-polynomial lower bound for the permanent

I Enough to bound the number of integer roots Adelic τ-conjecture [Phillipson & Rojas, 2012]

I Case k = 1: Follows from Descartes’ rule.I Case k = 2: Open.I Toy question: Number of real roots of fg + 1?

5 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 23: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Let’s make it real!Real τ-conjecture (Koiran, 2011)Let f =

∑ki=1

∏mj=1 fij where the fij ’s are t-sparse polynomials.

Then f has ≤ poly(k ,m, t) real roots.

Theorem (Koiran, 2011)Real τ-conjecture

=⇒ Super-polynomial lower bound for the permanent

I Enough to bound the number of integer roots

Adelic τ-conjecture [Phillipson & Rojas, 2012]

I Case k = 1: Follows from Descartes’ rule.I Case k = 2: Open.I Toy question: Number of real roots of fg + 1?

5 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 24: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Let’s make it real!Real τ-conjecture (Koiran, 2011)Let f =

∑ki=1

∏mj=1 fij where the fij ’s are t-sparse polynomials.

Then f has ≤ poly(k ,m, t) real roots.

Theorem (Koiran, 2011)Real τ-conjecture

=⇒ Super-polynomial lower bound for the permanent

I Enough to bound the number of integer roots Adelic τ-conjecture [Phillipson & Rojas, 2012]

I Case k = 1: Follows from Descartes’ rule.I Case k = 2: Open.I Toy question: Number of real roots of fg + 1?

5 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 25: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Let’s make it real!Real τ-conjecture (Koiran, 2011)Let f =

∑ki=1

∏mj=1 fij where the fij ’s are t-sparse polynomials.

Then f has ≤ poly(k ,m, t) real roots.

Theorem (Koiran, 2011)Real τ-conjecture

=⇒ Super-polynomial lower bound for the permanent

I Enough to bound the number of integer roots Adelic τ-conjecture [Phillipson & Rojas, 2012]

I Case k = 1: Follows from Descartes’ rule.

I Case k = 2: Open.I Toy question: Number of real roots of fg + 1?

5 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 26: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Let’s make it real!Real τ-conjecture (Koiran, 2011)Let f =

∑ki=1

∏mj=1 fij where the fij ’s are t-sparse polynomials.

Then f has ≤ poly(k ,m, t) real roots.

Theorem (Koiran, 2011)Real τ-conjecture

=⇒ Super-polynomial lower bound for the permanent

I Enough to bound the number of integer roots Adelic τ-conjecture [Phillipson & Rojas, 2012]

I Case k = 1: Follows from Descartes’ rule.I Case k = 2: Open.

I Toy question: Number of real roots of fg + 1?

5 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 27: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Let’s make it real!Real τ-conjecture (Koiran, 2011)Let f =

∑ki=1

∏mj=1 fij where the fij ’s are t-sparse polynomials.

Then f has ≤ poly(k ,m, t) real roots.

Theorem (Koiran, 2011)Real τ-conjecture

=⇒ Super-polynomial lower bound for the permanent

I Enough to bound the number of integer roots Adelic τ-conjecture [Phillipson & Rojas, 2012]

I Case k = 1: Follows from Descartes’ rule.I Case k = 2: Open.I Toy question: Number of real roots of fg + 1?

5 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 28: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Descartes’ rule without signs

TheoremIf f ∈ R[X ] has t monomials, then it has ≤ (t − 1) positive realroots.

Proof. Induction on t .

I t = 1: No positive real rootI t > 1: Let cαXα = lowest degree monomial.

• g = f /Xα: same positive roots, nonzero constant coefficient• g ′ has (t − 1) monomials =⇒ ≤ (t − 2) positive roots• There is a root of g ′ between two consecutive roots of g (Rolle’s

theorem)

f =∑k

i=1

∏mj=1 fij : ≤ 2ktm − 1 real roots

6 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 29: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Descartes’ rule without signs

TheoremIf f ∈ R[X ] has t monomials, then it has ≤ (t − 1) positive realroots.

Proof. Induction on t .I t = 1: No positive real root

I t > 1: Let cαXα = lowest degree monomial.

• g = f /Xα: same positive roots, nonzero constant coefficient• g ′ has (t − 1) monomials =⇒ ≤ (t − 2) positive roots• There is a root of g ′ between two consecutive roots of g (Rolle’s

theorem)

f =∑k

i=1

∏mj=1 fij : ≤ 2ktm − 1 real roots

6 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 30: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Descartes’ rule without signs

TheoremIf f ∈ R[X ] has t monomials, then it has ≤ (t − 1) positive realroots.

Proof. Induction on t .I t = 1: No positive real rootI t > 1: Let cαXα = lowest degree monomial.

• g = f /Xα: same positive roots, nonzero constant coefficient• g ′ has (t − 1) monomials =⇒ ≤ (t − 2) positive roots• There is a root of g ′ between two consecutive roots of g (Rolle’s

theorem)

f =∑k

i=1

∏mj=1 fij : ≤ 2ktm − 1 real roots

6 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 31: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Descartes’ rule without signs

TheoremIf f ∈ R[X ] has t monomials, then it has ≤ (t − 1) positive realroots.

Proof. Induction on t .I t = 1: No positive real rootI t > 1: Let cαXα = lowest degree monomial.

• g = f /Xα: same positive roots, nonzero constant coefficient

• g ′ has (t − 1) monomials =⇒ ≤ (t − 2) positive roots• There is a root of g ′ between two consecutive roots of g (Rolle’s

theorem)

f =∑k

i=1

∏mj=1 fij : ≤ 2ktm − 1 real roots

6 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 32: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Descartes’ rule without signs

TheoremIf f ∈ R[X ] has t monomials, then it has ≤ (t − 1) positive realroots.

Proof. Induction on t .I t = 1: No positive real rootI t > 1: Let cαXα = lowest degree monomial.

• g = f /Xα: same positive roots, nonzero constant coefficient• g ′ has (t − 1) monomials =⇒ ≤ (t − 2) positive roots

• There is a root of g ′ between two consecutive roots of g (Rolle’stheorem)

f =∑k

i=1

∏mj=1 fij : ≤ 2ktm − 1 real roots

6 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 33: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Descartes’ rule without signs

TheoremIf f ∈ R[X ] has t monomials, then it has ≤ (t − 1) positive realroots.

Proof. Induction on t .I t = 1: No positive real rootI t > 1: Let cαXα = lowest degree monomial.

• g = f /Xα: same positive roots, nonzero constant coefficient• g ′ has (t − 1) monomials =⇒ ≤ (t − 2) positive roots• There is a root of g ′ between two consecutive roots of g (Rolle’s

theorem)

f =∑k

i=1

∏mj=1 fij : ≤ 2ktm − 1 real roots

6 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 34: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Descartes’ rule without signs

TheoremIf f ∈ R[X ] has t monomials, then it has ≤ (t − 1) positive realroots.

Proof. Induction on t .I t = 1: No positive real rootI t > 1: Let cαXα = lowest degree monomial.

• g = f /Xα: same positive roots, nonzero constant coefficient• g ′ has (t − 1) monomials =⇒ ≤ (t − 2) positive roots• There is a root of g ′ between two consecutive roots of g (Rolle’s

theorem)

f =∑k

i=1

∏mj=1 fij : ≤ 2ktm − 1 real roots

6 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 35: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Real τ-conjecture =⇒ Permanent is hard

SPS(k ,m, t) =

f =k∑

i=1

m∏j=1

fij : fij ’s are t-sparse

Incorrect proof. Assume the permanent is easy.

I∏2n

i=1(X − i) has circuits of size poly(n) [Burgisser, 2007-09]I Reduction to depth 4 SPS polynomial of size 2o(n)

[Agrawal-Vinay, 2008]I Contradiction with real τ-conjecture

+ other details. . .

7 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 36: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Real τ-conjecture =⇒ Permanent is hard

SPS(k ,m, t) =

f =k∑

i=1

m∏j=1

fij : fij ’s are t-sparse

Incorrect proof. Assume the permanent is easy.

I∏2n

i=1(X − i) has circuits of size poly(n) [Burgisser, 2007-09]I Reduction to depth 4 SPS polynomial of size 2o(n)

[Agrawal-Vinay, 2008]I Contradiction with real τ-conjecture

+ other details. . .

7 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 37: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Real τ-conjecture =⇒ Permanent is hard

SPS(k ,m, t) =

f =k∑

i=1

m∏j=1

fij : fij ’s are t-sparse

Incorrect proof. Assume the permanent is easy.I∏2n

i=1(X − i) has circuits of size poly(n) [Burgisser, 2007-09]

I Reduction to depth 4 SPS polynomial of size 2o(n)

[Agrawal-Vinay, 2008]I Contradiction with real τ-conjecture

+ other details. . .

7 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 38: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Real τ-conjecture =⇒ Permanent is hard

SPS(k ,m, t) =

f =k∑

i=1

m∏j=1

fij : fij ’s are t-sparse

Incorrect proof. Assume the permanent is easy.I∏2n

i=1(X − i) has circuits of size poly(n) [Burgisser, 2007-09]I Reduction to depth 4 SPS polynomial of size 2o(n)

[Agrawal-Vinay, 2008]

I Contradiction with real τ-conjecture

+ other details. . .

7 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 39: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Real τ-conjecture =⇒ Permanent is hard

SPS(k ,m, t) =

f =k∑

i=1

m∏j=1

fij : fij ’s are t-sparse

Incorrect proof. Assume the permanent is easy.I∏2n

i=1(X − i) has circuits of size poly(n) [Burgisser, 2007-09]I Reduction to depth 4 SPS polynomial of size 2o(n)

[Agrawal-Vinay, 2008]I Contradiction with real τ-conjecture

+ other details. . .

7 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 40: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Real τ-conjecture =⇒ Permanent is hard

SPS(k ,m, t) =

f =k∑

i=1

m∏j=1

fij : fij ’s are t-sparse

Incorrect proof. Assume the permanent is easy.I∏2n

i=1(X − i) has circuits of size poly(n) [Burgisser, 2007-09]I Reduction to depth 4 SPS polynomial of size 2o(n)

[Agrawal-Vinay, 2008]I Contradiction with real τ-conjecture

+ other details. . .

7 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 41: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Real τ-conjecture =⇒ Permanent is hard

SPS(k ,m, t) =

f =k∑

i=1

m∏j=1

fij : fij ’s are t-sparse

In

correct proof. Assume the permanent is easy.I∏2n

i=1(X − i) has circuits of size poly(n) [Burgisser, 2007-09]I Reduction to depth 4 SPS polynomial of size 2o(n)

[Koiran, 2011]I Contradiction with real τ-conjecture

+ other details. . .

7 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 42: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Reduction to depth 4

Theorem (Koiran, 2011)Circuit of size t and degree d

Depth-4 circuit of size tO(√d log d)

Proof idea.I Construct an equivalent Arithmetic Branching Program size t log 2d + 1, depth δ = 3d − 1 [Malod-Portier, 2008]

I ABP ≡ Matrix powering

8 / 11The real τ-conjecture & lower bounds for the permanent

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Page 43: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Reduction to depth 4

Theorem (Koiran, 2011)Circuit of size t and degree d

Depth-4 circuit of size tO(√d log d)

Proof idea.I Construct an equivalent Arithmetic Branching Program size t log 2d + 1, depth δ = 3d − 1 [Malod-Portier, 2008]

I ABP ≡ Matrix powering

(x + y)(y + z) + y(x + x)

sx

y

x

x

yt

z

8 / 11The real τ-conjecture & lower bounds for the permanent

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Page 44: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Reduction to depth 4

Theorem (Koiran, 2011)Circuit of size t and degree d

Depth-4 circuit of size tO(√d log d)

Proof idea.I Construct an equivalent Arithmetic Branching Program size t log 2d + 1, depth δ = 3d − 1 [Malod-Portier, 2008]

I ABP ≡ Matrix powering

(x + y)(y + z) + y(x + x)

sx

y

x

x

yt

z

8 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 45: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Reduction to depth 4

Theorem (Koiran, 2011)Circuit of size t and degree d

Depth-4 circuit of size tO(√d log d)

Proof idea.I Construct an equivalent Arithmetic Branching Program size t log 2d + 1, depth δ = 3d − 1 [Malod-Portier, 2008]

I ABP ≡ Matrix powering

I Mδ = (M√δ)

√δ

8 / 11The real τ-conjecture & lower bounds for the permanent

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Page 46: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Reduction to depth 4

Theorem (Koiran, 2011)Circuit of size t and degree d

Depth-4 circuit of size tO(√d log d)

Proof idea.I Construct an equivalent Arithmetic Branching Program size t log 2d + 1, depth δ = 3d − 1 [Malod-Portier, 2008]

I ABP ≡ Matrix powering

I Mδ = (M√δ)

√δ

8 / 11The real τ-conjecture & lower bounds for the permanent

N

Consequence. Replace poly(k ,m, t) by 2polylog(k,m,t).

Page 47: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

The limited power of powering

SPS(k ,m, t,A) =

k∑

i=1

m∏j=1

fαij

j : fj ’s are t-sparse, αij ≤ A

Theorem (G.-Koiran-Portier-Strozecki, 2011)If f ∈ SPS(k ,m, t,A), its number of real roots is at most

C ·[e ·(1 +

tm

2k−1 − 1

)]2k−1−1for some C .

I Independent of A.I If k and m are fixed, this is polynomial in t .

9 / 11The real τ-conjecture & lower bounds for the permanent

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Page 48: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

The limited power of powering

SPS(k ,m, t,A) =

k∑

i=1

m∏j=1

fαij

j : fj ’s are t-sparse, αij ≤ A

Theorem (G.-Koiran-Portier-Strozecki, 2011)If f ∈ SPS(k ,m, t,A), its number of real roots is at most

C ·[e ·(1 +

tm

2k−1 − 1

)]2k−1−1for some C .

I Independent of A.I If k and m are fixed, this is polynomial in t .

9 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 49: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

The limited power of powering

SPS(k ,m, t,A) =

k∑

i=1

m∏j=1

fαij

j : fj ’s are t-sparse, αij ≤ A

Theorem (G.-Koiran-Portier-Strozecki, 2011)If f ∈ SPS(k ,m, t,A), its number of real roots is at most

C ·[e ·(1 +

tm

2k−1 − 1

)]2k−1−1for some C .

I Independent of A.

I If k and m are fixed, this is polynomial in t .

9 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 50: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

The limited power of powering

SPS(k ,m, t,A) =

k∑

i=1

m∏j=1

fαij

j : fj ’s are t-sparse, αij ≤ A

Theorem (G.-Koiran-Portier-Strozecki, 2011)If f ∈ SPS(k ,m, t,A), its number of real roots is at most

C ·[e ·(1 +

tm

2k−1 − 1

)]2k−1−1for some C .

I Independent of A.I If k and m are fixed, this is polynomial in t .

9 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 51: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Case k = 2

PropositionThe polynomial

f =m∏j=1

fαj

j +m∏j=1

fβjj

has at most 2mtm + 4m(t − 1) real roots.

Proof sketch. Let F = f /∏

j fαj

j = 1 +∏

j fβj−αj

j . Then

F ′ =m∏j=1

fβj−αj−1j︸ ︷︷ ︸

≤ 2m(t − 1) roots and poles

×m∑j=1

(βj − αj)f′j

∏l 6=j

fl︸ ︷︷ ︸≤ 2mtm − 1 roots

.

10 / 11The real τ-conjecture & lower bounds for the permanent

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Page 52: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Case k = 2

PropositionThe polynomial

f =m∏j=1

fαj

j +m∏j=1

fβjj

has at most 2mtm + 4m(t − 1) real roots.

Proof sketch. Let F = f /∏

j fαj

j = 1 +∏

j fβj−αj

j .

Then

F ′ =m∏j=1

fβj−αj−1j︸ ︷︷ ︸

≤ 2m(t − 1) roots and poles

×m∑j=1

(βj − αj)f′j

∏l 6=j

fl︸ ︷︷ ︸≤ 2mtm − 1 roots

.

10 / 11The real τ-conjecture & lower bounds for the permanent

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Page 53: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Case k = 2

PropositionThe polynomial

f =m∏j=1

fαj

j +m∏j=1

fβjj

has at most 2mtm + 4m(t − 1) real roots.

Proof sketch. Let F = f /∏

j fαj

j = 1 +∏

j fβj−αj

j . Then

F ′ =m∏j=1

fβj−αj−1j︸ ︷︷ ︸

≤ 2m(t − 1) roots and poles

×m∑j=1

(βj − αj)f′j

∏l 6=j

fl︸ ︷︷ ︸≤ 2mtm − 1 roots

.

10 / 11The real τ-conjecture & lower bounds for the permanent

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Page 54: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Conclusion

I Real τ-conjecture =⇒ VP0 6= VNP0

I Use your favorite real analysis tools!I Related:

• Adelic formulation: number of p-adic roots [Phillipson-Rojas]• Random fij : Work in progress [Briquel-Burgisser]• Consequences on repartition of complex roots [Hrubes]

Embarrassing Open ProblemLet f , g be t-sparse polynomials.

What is the maximum number real of roots of fg + 1?

Thank you for your attention!

11 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 55: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Conclusion

I Real τ-conjecture =⇒ VP0 6= VNP0

I Use your favorite real analysis tools!

I Related:

• Adelic formulation: number of p-adic roots [Phillipson-Rojas]• Random fij : Work in progress [Briquel-Burgisser]• Consequences on repartition of complex roots [Hrubes]

Embarrassing Open ProblemLet f , g be t-sparse polynomials.

What is the maximum number real of roots of fg + 1?

Thank you for your attention!

11 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 56: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Conclusion

I Real τ-conjecture =⇒ VP0 6= VNP0

I Use your favorite real analysis tools!I Related:

• Adelic formulation: number of p-adic roots [Phillipson-Rojas]• Random fij : Work in progress [Briquel-Burgisser]• Consequences on repartition of complex roots [Hrubes]

Embarrassing Open ProblemLet f , g be t-sparse polynomials.

What is the maximum number real of roots of fg + 1?

Thank you for your attention!

11 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 57: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Conclusion

I Real τ-conjecture =⇒ VP0 6= VNP0

I Use your favorite real analysis tools!I Related:

• Adelic formulation: number of p-adic roots [Phillipson-Rojas]

• Random fij : Work in progress [Briquel-Burgisser]• Consequences on repartition of complex roots [Hrubes]

Embarrassing Open ProblemLet f , g be t-sparse polynomials.

What is the maximum number real of roots of fg + 1?

Thank you for your attention!

11 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 58: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Conclusion

I Real τ-conjecture =⇒ VP0 6= VNP0

I Use your favorite real analysis tools!I Related:

• Adelic formulation: number of p-adic roots [Phillipson-Rojas]• Random fij : Work in progress [Briquel-Burgisser]

• Consequences on repartition of complex roots [Hrubes]

Embarrassing Open ProblemLet f , g be t-sparse polynomials.

What is the maximum number real of roots of fg + 1?

Thank you for your attention!

11 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 59: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Conclusion

I Real τ-conjecture =⇒ VP0 6= VNP0

I Use your favorite real analysis tools!I Related:

• Adelic formulation: number of p-adic roots [Phillipson-Rojas]• Random fij : Work in progress [Briquel-Burgisser]• Consequences on repartition of complex roots [Hrubes]

Embarrassing Open ProblemLet f , g be t-sparse polynomials.

What is the maximum number real of roots of fg + 1?

Thank you for your attention!

11 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 60: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Conclusion

I Real τ-conjecture =⇒ VP0 6= VNP0

I Use your favorite real analysis tools!I Related:

• Adelic formulation: number of p-adic roots [Phillipson-Rojas]• Random fij : Work in progress [Briquel-Burgisser]• Consequences on repartition of complex roots [Hrubes]

Embarrassing Open ProblemLet f , g be t-sparse polynomials.

What is the maximum number real of roots of fg + 1?

Thank you for your attention!

11 / 11The real τ-conjecture & lower bounds for the permanent

N

Page 61: The real -conjecture & lower bounds for the permanentgrenet/publis/talk_CoA12.pdf · Rencontres CoA 22 novembre 2012. Arithmetic Circuits f(x;y;z) = x4 +4x3y +6x2y2 +4xy3 +x2z +2xyz

Conclusion

I Real τ-conjecture =⇒ VP0 6= VNP0

I Use your favorite real analysis tools!I Related:

• Adelic formulation: number of p-adic roots [Phillipson-Rojas]• Random fij : Work in progress [Briquel-Burgisser]• Consequences on repartition of complex roots [Hrubes]

Embarrassing Open ProblemLet f , g be t-sparse polynomials.

What is the maximum number real of roots of fg + 1?

Thank you for your attention!

11 / 11The real τ-conjecture & lower bounds for the permanent

N