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### Transcript of The real -conjecture & lower bounds for the grenet/publis/talk_CoA12.pdf · Rencontres CoA 22...

• The real -conjecture&

lower bounds for the permanent

Bruno GrenetLIP ENS de Lyon

Rencontres CoA 22 novembre 2012

• 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 polynomial(f ) = size of its smallest circuit

representation

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

N

• 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 polynomial(f ) = size of its smallest circuit

representation

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

N

• 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 polynomial(f ) = size of its smallest circuit

representation

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

N

• 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 polynomial(f ) = size of its smallest circuit

representation

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

N

• 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 polynomial(f ) = size of its smallest circuit

representation

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

N

• 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 polynomial(f ) = size of its smallest circuit

representation

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

N

• 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 polynomial(f ) = size of its smallest circuit

representation

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

N

• 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 polynomial(f ) = size of its smallest circuit

representation

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

N

• 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 polynomial(f ) = size of its smallest circuit

representation

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

N

• 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 polynomial(f ) = size of its smallest circuit

representation

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

N

• 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 polynomial(f ) = size of its smallest circuit

representation

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

N

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

Theorem (Burgisser, 2007)-conjecture

= super-polynomial lower bound for the permanent

= (PERn) is not polynomially bounded in n

= VP0 6= VNP0

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

x11 x1n... ...xn1 xnn

= Sn

ni=1

xi(i)

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

N

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

Theorem (Burgisser, 2007)-conjecture

= super-polynomial lower bound for the permanent

= (PERn) is not polynomially bounded in n

= VP0 6= VNP0

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

x11 x1n... ...xn1 xnn

= Sn

ni=1

xi(i)

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

N

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

Theorem (Burgisser, 2007)-conjecture

= super-polynomial lower bound for the permanent

= (PERn) is not polynomially bounded in n

= VP0 6= VNP0

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

x11 x1n... ...xn1 xnn

= Sn

ni=1

xi(i)

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

N

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

Theorem (Burgisser, 2007)-conjecture

= super-polynomial lower bound for the permanent

= (PERn) is not polynomially bounded in n

= VP0 6= VNP0

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

x11 x1n... ...xn1 xnn

= Sn

ni=1

xi(i)

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

N

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

Theorem (Burgisser, 2007)-conjecture

= super-polynomial lower bound for the permanent

= (PERn) is not polynomially bounded in n

= VP0 6= VNP0

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

x11 x1n... ...xn1 xnn

= Sn

ni=1

xi(i)

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

N

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

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

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

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

N

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

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

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

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

N

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

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

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

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

N

• Lets make it real!Real -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 -conjecture

= Super-polynomial lower bound for the permanent

I Enough to bound the number of integer roots Adelic -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 -conjecture & lower bounds for the permanent

N

• Lets make it real!Real -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 -conjecture

= Super-polynomial lower bound for the permanent

I Enough to bound the number of integer roots Adelic -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 -conjecture & lower bounds for the permanent

N

• Lets make it real!Real -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 -conjecture

= Super-polynomial lower bound for the permanent

I Enough to bound the number of integer roots

Adelic -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 -conjecture & lower bounds for the permanent

N

• Lets make it real!Real -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 -conjecture

= Super-polynomial lower bound for the permanent

I Enough to bound the number of integer roots Adelic -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 -conjecture & lower bounds for the permanent

N

• Lets make it real!Real -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 -conjecture

= Super-polynomial lower bound for the permanent

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

I Case k = 1: Follows from Descartes rule.

I Case k = 2: Open.I T