Volume of alcoved polyhedra and Mahler conjecture M.J. de la Puente - - PowerPoint PPT Presentation

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Volume of alcoved polyhedra and Mahler conjecture M.J. de la Puente - - PowerPoint PPT Presentation

May 06, 2023 451 likes •2.04k views

Volume of alcoved polyhedra and Mahler conjecture M.J. de la Puente F. Matem aticas, U. Complutense (UCM), Madrid (Spain) mpuente@ucm.es (joint work with P.L. Claver a ) ISSAC 2018, CUNY, New York (USA) Vol. alc. polyhedr. Mahler conj.

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SLIDE 1

Volume of alcoved polyhedra and Mahler conjecture

M.J. de la Puente

  • F. Matem´

aticas, U. Complutense (UCM), Madrid (Spain) mpuente@ucm.es (joint work with P.L. Claver´ ıa) ISSAC 2018, CUNY, New York (USA)

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 1/22

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SLIDE 2

P ⊂ Rn body if P is compact convex, non–empty interior

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 2/22

slide-3
SLIDE 3

P ⊂ Rn body if P is compact convex, non–empty interior P centrally symmetric if P = −P

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 2/22

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SLIDE 4

P ⊂ Rn body if P is compact convex, non–empty interior P centrally symmetric if P = −P polar of P is P◦ := {x ∈ Rn : x, y ≤ 1, ∀y ∈ P}

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 2/22

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SLIDE 5

P ⊂ Rn body if P is compact convex, non–empty interior P centrally symmetric if P = −P polar of P is P◦ := {x ∈ Rn : x, y ≤ 1, ∀y ∈ P} polytope is convex hull in Rn of finite set

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 2/22

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SLIDE 6

P ⊂ Rn body if P is compact convex, non–empty interior P centrally symmetric if P = −P polar of P is P◦ := {x ∈ Rn : x, y ≤ 1, ∀y ∈ P} polytope is convex hull in Rn of finite set p1, p2, . . . , pr ∈ Rn,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 2/22

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SLIDE 7

P ⊂ Rn body if P is compact convex, non–empty interior P centrally symmetric if P = −P polar of P is P◦ := {x ∈ Rn : x, y ≤ 1, ∀y ∈ P} polytope is convex hull in Rn of finite set p1, p2, . . . , pr ∈ Rn, P = conv(p1, p2, . . . , pr)

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 2/22

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SLIDE 8

P = conv(p1, p2, . . . , pr),

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 3/22

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SLIDE 9

P = conv(p1, p2, . . . , pr), P◦ =

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 3/22

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SLIDE 10

P = conv(p1, p2, . . . , pr), P◦ = {x ∈ Rn : x, pk ≤ 1,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 3/22

slide-11
SLIDE 11

P = conv(p1, p2, . . . , pr), P◦ = {x ∈ Rn : x, pk ≤ 1, ∀k = 1, 2, . . . , r},

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 3/22

slide-12
SLIDE 12

P = conv(p1, p2, . . . , pr), P◦ = {x ∈ Rn : x, pk ≤ 1, ∀k = 1, 2, . . . , r}, Example:

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 3/22

slide-13
SLIDE 13

P = conv(p1, p2, . . . , pr), P◦ = {x ∈ Rn : x, pk ≤ 1, ∀k = 1, 2, . . . , r}, Example: p1 = (1, 1),

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 3/22

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SLIDE 14

P = conv(p1, p2, . . . , pr), P◦ = {x ∈ Rn : x, pk ≤ 1, ∀k = 1, 2, . . . , r}, Example: p1 = (1, 1), p2 = (1, −1),

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 3/22

slide-15
SLIDE 15

P = conv(p1, p2, . . . , pr), P◦ = {x ∈ Rn : x, pk ≤ 1, ∀k = 1, 2, . . . , r}, Example: p1 = (1, 1), p2 = (1, −1), p3 = −p1,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 3/22

slide-16
SLIDE 16

P = conv(p1, p2, . . . , pr), P◦ = {x ∈ Rn : x, pk ≤ 1, ∀k = 1, 2, . . . , r}, Example: p1 = (1, 1), p2 = (1, −1), p3 = −p1, p4 = −p2 ∈ R2,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 3/22

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SLIDE 17

P = conv(p1, p2, . . . , pr), P◦ = {x ∈ Rn : x, pk ≤ 1, ∀k = 1, 2, . . . , r}, Example: p1 = (1, 1), p2 = (1, −1), p3 = −p1, p4 = −p2 ∈ R2,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 3/22

slide-18
SLIDE 18

P = conv(p1, p2, . . . , pr), P◦ = {x ∈ Rn : x, pk ≤ 1, ∀k = 1, 2, . . . , r}, Example: p1 = (1, 1), p2 = (1, −1), p3 = −p1, p4 = −p2 ∈ R2, x, p1 = x1 + x2,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 3/22

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SLIDE 19

P = conv(p1, p2, . . . , pr), P◦ = {x ∈ Rn : x, pk ≤ 1, ∀k = 1, 2, . . . , r}, Example: p1 = (1, 1), p2 = (1, −1), p3 = −p1, p4 = −p2 ∈ R2, x, p1 = x1 + x2, x, p2 = x1 − x2,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 3/22

slide-20
SLIDE 20

P = conv(p1, p2, . . . , pr), P◦ = {x ∈ Rn : x, pk ≤ 1, ∀k = 1, 2, . . . , r}, Example: p1 = (1, 1), p2 = (1, −1), p3 = −p1, p4 = −p2 ∈ R2, x, p1 = x1 + x2, x, p2 = x1 − x2, x, p3 = −x1 − x2,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 3/22

slide-21
SLIDE 21

P = conv(p1, p2, . . . , pr), P◦ = {x ∈ Rn : x, pk ≤ 1, ∀k = 1, 2, . . . , r}, Example: p1 = (1, 1), p2 = (1, −1), p3 = −p1, p4 = −p2 ∈ R2, x, p1 = x1 + x2, x, p2 = x1 − x2, x, p3 = −x1 − x2, x, p4 = −x1 + x2

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 3/22

slide-22
SLIDE 22

P ⊂ Rn centrally symmetric convex body,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 4/22

slide-23
SLIDE 23

P ⊂ Rn centrally symmetric convex body, Euclidean norm,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 4/22

slide-24
SLIDE 24

P ⊂ Rn centrally symmetric convex body, Euclidean norm, usual volume,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 4/22

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SLIDE 25

P ⊂ Rn centrally symmetric convex body, Euclidean norm, usual volume, Key obs: When vol(P) increases, then vol(P◦) decreases

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 4/22

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SLIDE 26

P ⊂ Rn centrally symmetric convex body, Euclidean norm, usual volume, Key obs: When vol(P) increases, then vol(P◦) decreases

How large is

vol(P) vol(P◦)?

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 4/22

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SLIDE 27

P ⊂ Rn centrally symmetric convex body, Euclidean norm, usual volume, Key obs: When vol(P) increases, then vol(P◦) decreases

How large is

vol(P) vol(P◦)? Unit cube Cn (edge= 2),

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 4/22

slide-28
SLIDE 28

P ⊂ Rn centrally symmetric convex body, Euclidean norm, usual volume, Key obs: When vol(P) increases, then vol(P◦) decreases

How large is

vol(P) vol(P◦)? Unit cube Cn (edge= 2), C ◦

n is unit cross–polytope

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 4/22

slide-29
SLIDE 29

P ⊂ Rn centrally symmetric convex body, Euclidean norm, usual volume, Key obs: When vol(P) increases, then vol(P◦) decreases

How large is

vol(P) vol(P◦)? Unit cube Cn (edge= 2), C ◦

n is unit cross–polytope

Unit ball Bn

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 4/22

slide-30
SLIDE 30

P ⊂ Rn centrally symmetric convex body, Euclidean norm, usual volume, Key obs: When vol(P) increases, then vol(P◦) decreases

How large is

vol(P) vol(P◦)? Unit cube Cn (edge= 2), C ◦

n is unit cross–polytope

Unit ball Bn = B◦

n

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 4/22

slide-31
SLIDE 31

P ⊂ Rn centrally symmetric convex body, Euclidean norm, usual volume, Key obs: When vol(P) increases, then vol(P◦) decreases

How large is

vol(P) vol(P◦)? Unit cube Cn (edge= 2), C ◦

n is unit cross–polytope

Unit ball Bn = B◦

n

Blaschke–Santal´

  • inequality (proved 1949):
  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 4/22

slide-32
SLIDE 32

P ⊂ Rn centrally symmetric convex body, Euclidean norm, usual volume, Key obs: When vol(P) increases, then vol(P◦) decreases

How large is

vol(P) vol(P◦)? Unit cube Cn (edge= 2), C ◦

n is unit cross–polytope

Unit ball Bn = B◦

n

Blaschke–Santal´

  • inequality (proved 1949):

vol(P) vol(P◦) ≤ vol(Bn) vol(B◦

n)

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 4/22

slide-33
SLIDE 33

P ⊂ Rn centrally symmetric convex body, Euclidean norm, usual volume, Key obs: When vol(P) increases, then vol(P◦) decreases

How large is

vol(P) vol(P◦)? Unit cube Cn (edge= 2), C ◦

n is unit cross–polytope

Unit ball Bn = B◦

n

Blaschke–Santal´

  • inequality (proved 1949):

vol(P) vol(P◦) ≤ vol(Bn) vol(B◦

n)

Mahler conjecture (open since 1939):

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 4/22

slide-34
SLIDE 34

P ⊂ Rn centrally symmetric convex body, Euclidean norm, usual volume, Key obs: When vol(P) increases, then vol(P◦) decreases

How large is

vol(P) vol(P◦)? Unit cube Cn (edge= 2), C ◦

n is unit cross–polytope

Unit ball Bn = B◦

n

Blaschke–Santal´

  • inequality (proved 1949):

vol(P) vol(P◦) ≤ vol(Bn) vol(B◦

n)

Mahler conjecture (open since 1939): vol(Cn) vol(C ◦

n) ≤ vol(P) vol(P◦)

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 4/22

slide-35
SLIDE 35

P ⊂ Rn centrally symmetric convex body

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 5/22

slide-36
SLIDE 36

P ⊂ Rn centrally symmetric convex body Unit cube vol(Cn) = 2n,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 5/22

slide-37
SLIDE 37

P ⊂ Rn centrally symmetric convex body Unit cube vol(Cn) = 2n, vol(C ◦

n) = 2n n!

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 5/22

slide-38
SLIDE 38

P ⊂ Rn centrally symmetric convex body Unit cube vol(Cn) = 2n, vol(C ◦

n) = 2n n!

Unit ball vol(Bn) =

πn/2 Γ( n

2+1)

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 5/22

slide-39
SLIDE 39

P ⊂ Rn centrally symmetric convex body Unit cube vol(Cn) = 2n, vol(C ◦

n) = 2n n!

Unit ball vol(Bn) =

πn/2 Γ( n

2+1)

4n n!

?

≤ vol(P) vol(P◦) ≤ πn Γ(n

2 + 1)2

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 5/22

slide-40
SLIDE 40

P ⊂ Rn centrally symmetric convex body Unit cube vol(Cn) = 2n, vol(C ◦

n) = 2n n!

Unit ball vol(Bn) =

πn/2 Γ( n

2+1)

4n n!

?

≤ vol(P) vol(P◦) ≤ πn Γ(n

2 + 1)2

is Mahler conj.

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 5/22

slide-41
SLIDE 41

P ⊂ Rn centrally symmetric convex body Unit cube vol(Cn) = 2n, vol(C ◦

n) = 2n n!

Unit ball vol(Bn) =

πn/2 Γ( n

2+1)

4n n!

?

≤ vol(P) vol(P◦) ≤ πn Γ(n

2 + 1)2

is Mahler conj. and Blaschke–Santal´

  • thm.
  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 5/22

slide-42
SLIDE 42

To compute the volume of a polytope

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 6/22

slide-43
SLIDE 43

To compute the volume of a polytope is a #P–problem

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 6/22

slide-44
SLIDE 44

To compute the volume of a polytope is a #P–problem (Dyer and Frieze 1988, L. Khachiyan 1989)

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 6/22

slide-45
SLIDE 45

Alcoved polytope:

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

slide-46
SLIDE 46

Alcoved polytope: equations of facets of P ⊂ Rn are

  • f TWO types:
  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

slide-47
SLIDE 47

Alcoved polytope: equations of facets of P ⊂ Rn are

  • f TWO types:

xi = ai

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

slide-48
SLIDE 48

Alcoved polytope: equations of facets of P ⊂ Rn are

  • f TWO types:

xi = ai xi − xj = aij,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

slide-49
SLIDE 49

Alcoved polytope: equations of facets of P ⊂ Rn are

  • f TWO types:

xi = ai xi − xj = aij, with ai, aij ∈ R

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

slide-50
SLIDE 50

Alcoved polytope: equations of facets of P ⊂ Rn are

  • f TWO types:

xi = ai xi − xj = aij, with ai, aij ∈ R ⇒ arrange coeffs. in matrix (n + 1) × (n + 1) ⇒ do linear algebra

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

slide-51
SLIDE 51

Alcoved polytope: equations of facets of P ⊂ Rn are

  • f TWO types:

xi = ai xi − xj = aij, with ai, aij ∈ R ⇒ arrange coeffs. in matrix (n + 1) × (n + 1) ⇒ do linear algebra

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

slide-52
SLIDE 52

Alcoved polytope: equations of facets of P ⊂ Rn are

  • f TWO types:

xi = ai xi − xj = aij, with ai, aij ∈ R ⇒ arrange coeffs. in matrix (n + 1) × (n + 1) ⇒ do linear algebra A(P) =   0 −6 −6 −5 0 −4 −4 −3  

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

slide-53
SLIDE 53

Alcoved polytope: equations of facets of P ⊂ Rn are

  • f TWO types:

xi = ai xi − xj = aij, with ai, aij ∈ R ⇒ arrange coeffs. in matrix (n + 1) × (n + 1) ⇒ do linear algebra A(P) =   0 −6 −6 −5 0 −4 −4 −3   ai,n+1 ≤ xi ≤ −an+1,i

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

slide-54
SLIDE 54

Alcoved polytope: equations of facets of P ⊂ Rn are

  • f TWO types:

xi = ai xi − xj = aij, with ai, aij ∈ R ⇒ arrange coeffs. in matrix (n + 1) × (n + 1) ⇒ do linear algebra A(P) =   0 −6 −6 −5 0 −4 −4 −3   ai,n+1 ≤ xi ≤ −an+1,i ai,j ≤ xi − xj ≤ −aj,i

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

slide-55
SLIDE 55

Alcoved polytope: equations of facets of P ⊂ Rn are

  • f TWO types:

xi = ai xi − xj = aij, with ai, aij ∈ R ⇒ arrange coeffs. in matrix (n + 1) × (n + 1) ⇒ do linear algebra A(P) =   0 −6 −6 −5 0 −4 −4 −3   ai,n+1 ≤ xi ≤ −an+1,i ai,j ≤ xi − xj ≤ −aj,i aii = 0

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 7/22

slide-56
SLIDE 56

Which matrices [aij] ∈ Mn+1 yield alcoved polytopes P ⊂ Rn?

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

slide-57
SLIDE 57

Which matrices [aij] ∈ Mn+1 yield alcoved polytopes P ⊂ Rn?

  • Thm. (Sergeev 2009, de la Puente 2013)
  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

slide-58
SLIDE 58

Which matrices [aij] ∈ Mn+1 yield alcoved polytopes P ⊂ Rn?

  • Thm. (Sergeev 2009, de la Puente 2013) Every normal

idempotent matrix gives rise to an alcoved polytope

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

slide-59
SLIDE 59

Which matrices [aij] ∈ Mn+1 yield alcoved polytopes P ⊂ Rn?

  • Thm. (Sergeev 2009, de la Puente 2013) Every normal

idempotent matrix gives rise to an alcoved polytope A = [aij] normal matrix: aii = 0 and aij ≤ 0, all i, j

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

slide-60
SLIDE 60

Which matrices [aij] ∈ Mn+1 yield alcoved polytopes P ⊂ Rn?

  • Thm. (Sergeev 2009, de la Puente 2013) Every normal

idempotent matrix gives rise to an alcoved polytope A = [aij] normal matrix: aii = 0 and aij ≤ 0, all i, j A idempotent (wrt tropical product): A ⊙ A = A

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

slide-61
SLIDE 61

Which matrices [aij] ∈ Mn+1 yield alcoved polytopes P ⊂ Rn?

  • Thm. (Sergeev 2009, de la Puente 2013) Every normal

idempotent matrix gives rise to an alcoved polytope A = [aij] normal matrix: aii = 0 and aij ≤ 0, all i, j A idempotent (wrt tropical product): A ⊙ A = A ⊕ = max

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

slide-62
SLIDE 62

Which matrices [aij] ∈ Mn+1 yield alcoved polytopes P ⊂ Rn?

  • Thm. (Sergeev 2009, de la Puente 2013) Every normal

idempotent matrix gives rise to an alcoved polytope A = [aij] normal matrix: aii = 0 and aij ≤ 0, all i, j A idempotent (wrt tropical product): A ⊙ A = A ⊕ = max is trop. sum

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

slide-63
SLIDE 63

Which matrices [aij] ∈ Mn+1 yield alcoved polytopes P ⊂ Rn?

  • Thm. (Sergeev 2009, de la Puente 2013) Every normal

idempotent matrix gives rise to an alcoved polytope A = [aij] normal matrix: aii = 0 and aij ≤ 0, all i, j A idempotent (wrt tropical product): A ⊙ A = A ⊕ = max is trop. sum and ⊙ = +

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

slide-64
SLIDE 64

Which matrices [aij] ∈ Mn+1 yield alcoved polytopes P ⊂ Rn?

  • Thm. (Sergeev 2009, de la Puente 2013) Every normal

idempotent matrix gives rise to an alcoved polytope A = [aij] normal matrix: aii = 0 and aij ≤ 0, all i, j A idempotent (wrt tropical product): A ⊙ A = A ⊕ = max is trop. sum and ⊙ = + is trop. product

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 8/22

slide-65
SLIDE 65

⊕ = max tropical sum, ⊙ = + trop. product

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 9/22

slide-66
SLIDE 66

⊕ = max tropical sum, ⊙ = + trop. product A =   0 −6 −6 −5 0 −4 −4 −3   B =   0 −2 −7 −5 −5   A ⊕ B =   0 −2 −6 −4 −3   A ⊙ B =   0 −2 −6 max{−5, 0, −9} max{−4, −3, −5} −3   A, B are normal A idempotent,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 9/22

slide-67
SLIDE 67

⊕ = max tropical sum, ⊙ = + trop. product A =   0 −6 −6 −5 0 −4 −4 −3   B =   0 −2 −7 −5 −5   A ⊕ B =   0 −2 −6 −4 −3   A ⊙ B =   0 −2 −6 max{−5, 0, −9} max{−4, −3, −5} −3   A, B are normal A idempotent, B ⊙ B =   0 −2 −2 −5 −5  ,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 9/22

slide-68
SLIDE 68

⊕ = max tropical sum, ⊙ = + trop. product A =   0 −6 −6 −5 0 −4 −4 −3   B =   0 −2 −7 −5 −5   A ⊕ B =   0 −2 −6 −4 −3   A ⊙ B =   0 −2 −6 max{−5, 0, −9} max{−4, −3, −5} −3   A, B are normal A idempotent, B ⊙ B =   0 −2 −2 −5 −5  , B not idempotent

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 9/22

slide-69
SLIDE 69

⊕ = max tropical sum, ⊙ = + trop. product A =   0 −6 −6 −5 0 −4 −4 −3   B =   0 −2 −7 −5 −5   A ⊕ B =   0 −2 −6 −4 −3   A ⊙ B =   0 −2 −6 max{−5, 0, −9} max{−4, −3, −5} −3   A, B are normal A idempotent, B ⊙ B =   0 −2 −2 −5 −5  , B not idempotent P(A) alcoved,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 9/22

slide-70
SLIDE 70

⊕ = max tropical sum, ⊙ = + trop. product A =   0 −6 −6 −5 0 −4 −4 −3   B =   0 −2 −7 −5 −5   A ⊕ B =   0 −2 −6 −4 −3   A ⊙ B =   0 −2 −6 max{−5, 0, −9} max{−4, −3, −5} −3   A, B are normal A idempotent, B ⊙ B =   0 −2 −2 −5 −5  , B not idempotent P(A) alcoved, P(B) not alcoved

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 9/22

slide-71
SLIDE 71

P(A) alcoved (hence convex), P(B) not alcoved

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 10/22

slide-72
SLIDE 72

dodecahedron with f –vector (v, e, f ) = (20, 30, 12)

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 11/22

slide-73
SLIDE 73

For n = 4 get volume of alcoved P ⊂ R3

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

slide-74
SLIDE 74

For n = 4 get volume of alcoved P ⊂ R3 in terms of matrix entries aij of A ∈ M4?

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

slide-75
SLIDE 75

For n = 4 get volume of alcoved P ⊂ R3 in terms of matrix entries aij of A ∈ M4?

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

slide-76
SLIDE 76

For n = 4 get volume of alcoved P ⊂ R3 in terms of matrix entries aij of A ∈ M4? normal idempotent A = B − E

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

slide-77
SLIDE 77

For n = 4 get volume of alcoved P ⊂ R3 in terms of matrix entries aij of A ∈ M4? normal idempotent A = B − E define bounding box B

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

slide-78
SLIDE 78

For n = 4 get volume of alcoved P ⊂ R3 in terms of matrix entries aij of A ∈ M4? normal idempotent A = B − E define bounding box B

  • def. six cants
  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

slide-79
SLIDE 79

For n = 4 get volume of alcoved P ⊂ R3 in terms of matrix entries aij of A ∈ M4? normal idempotent A = B − E define bounding box B

  • def. six cants arranged in cycle
  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

slide-80
SLIDE 80

For n = 4 get volume of alcoved P ⊂ R3 in terms of matrix entries aij of A ∈ M4? normal idempotent A = B − E define bounding box B

  • def. six cants arranged in cycle

Easy: vol(P) = vol(box)−6

j=1 vol Pj + j vol(Pj ∩Pj+1)

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

slide-81
SLIDE 81

For n = 4 get volume of alcoved P ⊂ R3 in terms of matrix entries aij of A ∈ M4? normal idempotent A = B − E define bounding box B

  • def. six cants arranged in cycle

Easy: vol(P) = vol(box)−6

j=1 vol Pj + j vol(Pj ∩Pj+1)

Thm.:

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

slide-82
SLIDE 82

For n = 4 get volume of alcoved P ⊂ R3 in terms of matrix entries aij of A ∈ M4? normal idempotent A = B − E define bounding box B

  • def. six cants arranged in cycle

Easy: vol(P) = vol(box)−6

j=1 vol Pj + j vol(Pj ∩Pj+1)

Thm.: vol(P) = ℓ1ℓ2ℓ3 −

6

  • j=1

c2

j ℓj

2 +

6

  • j=1

m2

j Mj

2 − m3

j

6

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

slide-83
SLIDE 83

For n = 4 get volume of alcoved P ⊂ R3 in terms of matrix entries aij of A ∈ M4? normal idempotent A = B − E define bounding box B

  • def. six cants arranged in cycle

Easy: vol(P) = vol(box)−6

j=1 vol Pj + j vol(Pj ∩Pj+1)

Thm.: vol(P) = ℓ1ℓ2ℓ3 −

6

  • j=1

c2

j ℓj

2 +

6

  • j=1

m2

j Mj

2 − m3

j

6 ℓ1, ℓ2, ℓ3 are edge–lengths of bounding box B

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

slide-84
SLIDE 84

For n = 4 get volume of alcoved P ⊂ R3 in terms of matrix entries aij of A ∈ M4? normal idempotent A = B − E define bounding box B

  • def. six cants arranged in cycle

Easy: vol(P) = vol(box)−6

j=1 vol Pj + j vol(Pj ∩Pj+1)

Thm.: vol(P) = ℓ1ℓ2ℓ3 −

6

  • j=1

c2

j ℓj

2 +

6

  • j=1

m2

j Mj

2 − m3

j

6 ℓ1, ℓ2, ℓ3 are edge–lengths of bounding box B mj := min{|cj|, |cj+1|},

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

slide-85
SLIDE 85

For n = 4 get volume of alcoved P ⊂ R3 in terms of matrix entries aij of A ∈ M4? normal idempotent A = B − E define bounding box B

  • def. six cants arranged in cycle

Easy: vol(P) = vol(box)−6

j=1 vol Pj + j vol(Pj ∩Pj+1)

Thm.: vol(P) = ℓ1ℓ2ℓ3 −

6

  • j=1

c2

j ℓj

2 +

6

  • j=1

m2

j Mj

2 − m3

j

6 ℓ1, ℓ2, ℓ3 are edge–lengths of bounding box B mj := min{|cj|, |cj+1|}, Mj := max{|cj|, |cj+1|}, cj is cant parameter, j = 1, 2, . . . , 6

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 12/22

slide-86
SLIDE 86
  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 13/22

slide-87
SLIDE 87
  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 13/22

slide-88
SLIDE 88
  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 13/22

slide-89
SLIDE 89

vol(P) = ℓ1ℓ2ℓ3 −

6

  • j=1

c2

j ℓj

2 +

6

  • j=1

m2

j Mj

2 − m3

j

6

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

slide-90
SLIDE 90

vol(P) = ℓ1ℓ2ℓ3 −

6

  • j=1

c2

j ℓj

2 +

6

  • j=1

m2

j Mj

2 − m3

j

6 ℓk = ak4, k = 1, 2, 3

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

slide-91
SLIDE 91

vol(P) = ℓ1ℓ2ℓ3 −

6

  • j=1

c2

j ℓj

2 +

6

  • j=1

m2

j Mj

2 − m3

j

6 ℓk = ak4, k = 1, 2, 3 each cj cant parameter is the difference of two entries in A

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

slide-92
SLIDE 92

vol(P) = ℓ1ℓ2ℓ3 −

6

  • j=1

c2

j ℓj

2 +

6

  • j=1

m2

j Mj

2 − m3

j

6 ℓk = ak4, k = 1, 2, 3 each cj cant parameter is the difference of two entries in A mj := min{|cj|, |cj+1|},

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

slide-93
SLIDE 93

vol(P) = ℓ1ℓ2ℓ3 −

6

  • j=1

c2

j ℓj

2 +

6

  • j=1

m2

j Mj

2 − m3

j

6 ℓk = ak4, k = 1, 2, 3 each cj cant parameter is the difference of two entries in A mj := min{|cj|, |cj+1|}, Mj := max{|cj|, |cj+1|},

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

slide-94
SLIDE 94

vol(P) = ℓ1ℓ2ℓ3 −

6

  • j=1

c2

j ℓj

2 +

6

  • j=1

m2

j Mj

2 − m3

j

6 ℓk = ak4, k = 1, 2, 3 each cj cant parameter is the difference of two entries in A mj := min{|cj|, |cj+1|}, Mj := max{|cj|, |cj+1|}, vol(P) is deg 3 rational homog polynomial in aij

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

slide-95
SLIDE 95

vol(P) = ℓ1ℓ2ℓ3 −

6

  • j=1

c2

j ℓj

2 +

6

  • j=1

m2

j Mj

2 − m3

j

6 ℓk = ak4, k = 1, 2, 3 each cj cant parameter is the difference of two entries in A mj := min{|cj|, |cj+1|}, Mj := max{|cj|, |cj+1|}, vol(P) is deg 3 rational homog polynomial in aij A = B − E

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

slide-96
SLIDE 96

vol(P) = ℓ1ℓ2ℓ3 −

6

  • j=1

c2

j ℓj

2 +

6

  • j=1

m2

j Mj

2 − m3

j

6 ℓk = ak4, k = 1, 2, 3 each cj cant parameter is the difference of two entries in A mj := min{|cj|, |cj+1|}, Mj := max{|cj|, |cj+1|}, vol(P) is deg 3 rational homog polynomial in aij A = B − E ℓk are the dimensions of bounding box B

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

slide-97
SLIDE 97

vol(P) = ℓ1ℓ2ℓ3 −

6

  • j=1

c2

j ℓj

2 +

6

  • j=1

m2

j Mj

2 − m3

j

6 ℓk = ak4, k = 1, 2, 3 each cj cant parameter is the difference of two entries in A mj := min{|cj|, |cj+1|}, Mj := max{|cj|, |cj+1|}, vol(P) is deg 3 rational homog polynomial in aij A = B − E ℓk are the dimensions of bounding box B the entries of E are the cj’s or zero

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 14/22

slide-98
SLIDE 98

Mahler conjecture for 3–dim alcoved polytopes

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 15/22

slide-99
SLIDE 99

Mahler conjecture for 3–dim alcoved polytopes

  • Thm. (de la Puente, 2012)
  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 15/22

slide-100
SLIDE 100

Mahler conjecture for 3–dim alcoved polytopes

  • Thm. (de la Puente, 2012) A symmetric matrix

⇔ P centrally symmetric

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 15/22

slide-101
SLIDE 101

Mahler conjecture for 3–dim alcoved polytopes

  • Thm. (de la Puente, 2012) A symmetric matrix

⇔ P centrally symmetric Reduction by affine map to case

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 15/22

slide-102
SLIDE 102

Mahler conjecture for 3–dim alcoved polytopes

  • Thm. (de la Puente, 2012) A symmetric matrix

⇔ P centrally symmetric Reduction by affine map to case P is unit cube (edge= 2),

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 15/22

slide-103
SLIDE 103

Mahler conjecture for 3–dim alcoved polytopes

  • Thm. (de la Puente, 2012) A symmetric matrix

⇔ P centrally symmetric Reduction by affine map to case P is unit cube (edge= 2), and cant parameters are x, y, z

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 15/22

slide-104
SLIDE 104

Mahler conjecture for 3–dim alcoved polytopes

  • Thm. (de la Puente, 2012) A symmetric matrix

⇔ P centrally symmetric Reduction by affine map to case P is unit cube (edge= 2), and cant parameters are x, y, z with −1 ≤ z ≤ y ≤ x ≤ 0

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 15/22

slide-105
SLIDE 105

Mahler conjecture for 3–dim alcoved polytopes

  • Thm. (de la Puente, 2012) A symmetric matrix

⇔ P centrally symmetric Reduction by affine map to case P is unit cube (edge= 2), and cant parameters are x, y, z with −1 ≤ z ≤ y ≤ x ≤ 0 vol(P) = 8−x2 (y + z)−y 2z+ 1

3

  • 2x3 + y 3

−2

  • x2 + y 2 + z2
  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 15/22

slide-106
SLIDE 106

Mahler conjecture for 3–dim alcoved polytopes

  • Thm. (de la Puente, 2012) A symmetric matrix

⇔ P centrally symmetric Reduction by affine map to case P is unit cube (edge= 2), and cant parameters are x, y, z with −1 ≤ z ≤ y ≤ x ≤ 0 vol(P) = 8−x2 (y + z)−y 2z+ 1

3

  • 2x3 + y 3

−2

  • x2 + y 2 + z2

vol(P◦) =

2/3 (2+x)(2+y) + 2/3 (2+y)(2+z) + 2/3 (2+z)(2+x) + 1/3 2+y + 2/3 2+z + 1 3

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 15/22

slide-107
SLIDE 107

Multiply,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 16/22

slide-108
SLIDE 108

Multiply, clear denominators,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 16/22

slide-109
SLIDE 109

Multiply, clear denominators, pass terms to LHS, get deg 6 polynomial MC = 2x4yz − 3x3y 2z − 3x3yz2 + xy 4z − 3xy 3z2 + 8x4y + 6x4z − 12x3y 2 −23x3yz − 9x3z2 − 6x2y 2z − 6x2yz2 + 4xy 4 − 15xy 3z − 9xy 2z2 − 6xyz3 + 2y 4z − 6y 3z2 + 24x4 − 40x3y − 38x3z − 30x2y 2 −66x2yz − 24x2z2 − 12xy 3 − 54xy 2z − 24xyz2 − 18xz3 + 10y 4 − 34y 3z − 24y 2z2 − 12yz3 − 8x3 − 156x2y − 144x2z − 72xy 2 −72xyz −72xz2−28y 3−144y 2z −60yz2−48z3−192x2− 96xy −120xz−192y 2−144yz−192z2−96x−144y −192z

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 16/22

slide-110
SLIDE 110

Multiply, clear denominators, pass terms to LHS, get deg 6 polynomial MC = 2x4yz − 3x3y 2z − 3x3yz2 + xy 4z − 3xy 3z2 + 8x4y + 6x4z − 12x3y 2 −23x3yz − 9x3z2 − 6x2y 2z − 6x2yz2 + 4xy 4 − 15xy 3z − 9xy 2z2 − 6xyz3 + 2y 4z − 6y 3z2 + 24x4 − 40x3y − 38x3z − 30x2y 2 −66x2yz − 24x2z2 − 12xy 3 − 54xy 2z − 24xyz2 − 18xz3 + 10y 4 − 34y 3z − 24y 2z2 − 12yz3 − 8x3 − 156x2y − 144x2z − 72xy 2 −72xyz −72xz2−28y 3−144y 2z −60yz2−48z3−192x2− 96xy −120xz−192y 2−144yz−192z2−96x−144y −192z MC|S ≥ 0, S simplex − 1 ≤ z ≤ y ≤ x ≤ 0 Mahler c.

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 16/22

slide-111
SLIDE 111

Prove MC is non–negative on semi–algebraic set S

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-112
SLIDE 112

Prove MC is non–negative on semi–algebraic set S How?

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-113
SLIDE 113

Prove MC is non–negative on semi–algebraic set S How? Express MC as a linear comb., with non–negative coeffs., of products of polys wj, non–negative on S.

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-114
SLIDE 114

Prove MC is non–negative on semi–algebraic set S How? Express MC as a linear comb., with non–negative coeffs., of products of polys wj, non–negative on S. Which wj?

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-115
SLIDE 115

Prove MC is non–negative on semi–algebraic set S How? Express MC as a linear comb., with non–negative coeffs., of products of polys wj, non–negative on S. Which wj?

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-116
SLIDE 116

Prove MC is non–negative on semi–algebraic set S How? Express MC as a linear comb., with non–negative coeffs., of products of polys wj, non–negative on S. Which wj? w1 = 1 + z,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-117
SLIDE 117

Prove MC is non–negative on semi–algebraic set S How? Express MC as a linear comb., with non–negative coeffs., of products of polys wj, non–negative on S. Which wj? w1 = 1 + z, w2 = y − z,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-118
SLIDE 118

Prove MC is non–negative on semi–algebraic set S How? Express MC as a linear comb., with non–negative coeffs., of products of polys wj, non–negative on S. Which wj? w1 = 1 + z, w2 = y − z, w3 = x − y,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-119
SLIDE 119

Prove MC is non–negative on semi–algebraic set S How? Express MC as a linear comb., with non–negative coeffs., of products of polys wj, non–negative on S. Which wj? w1 = 1 + z, w2 = y − z, w3 = x − y, w4 = −x

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-120
SLIDE 120

Prove MC is non–negative on semi–algebraic set S How? Express MC as a linear comb., with non–negative coeffs., of products of polys wj, non–negative on S. Which wj? w1 = 1 + z, w2 = y − z, w3 = x − y, w4 = −x giving equations of the S facets

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-121
SLIDE 121

Prove MC is non–negative on semi–algebraic set S How? Express MC as a linear comb., with non–negative coeffs., of products of polys wj, non–negative on S. Which wj? w1 = 1 + z, w2 = y − z, w3 = x − y, w4 = −x giving equations of the S facets 1 = w1 + w2 + w3 + w4 (relation)

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-122
SLIDE 122

Prove MC is non–negative on semi–algebraic set S How? Express MC as a linear comb., with non–negative coeffs., of products of polys wj, non–negative on S. Which wj? w1 = 1 + z, w2 = y − z, w3 = x − y, w4 = −x giving equations of the S facets 1 = w1 + w2 + w3 + w4 (relation) Thus, compute MCj = 16−jMCj

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-123
SLIDE 123

Prove MC is non–negative on semi–algebraic set S How? Express MC as a linear comb., with non–negative coeffs., of products of polys wj, non–negative on S. Which wj? w1 = 1 + z, w2 = y − z, w3 = x − y, w4 = −x giving equations of the S facets 1 = w1 + w2 + w3 + w4 (relation) Thus, compute MCj = 16−jMCj = (w1 + w2 + w3 + w4)6−jMCj,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-124
SLIDE 124

Prove MC is non–negative on semi–algebraic set S How? Express MC as a linear comb., with non–negative coeffs., of products of polys wj, non–negative on S. Which wj? w1 = 1 + z, w2 = y − z, w3 = x − y, w4 = −x giving equations of the S facets 1 = w1 + w2 + w3 + w4 (relation) Thus, compute MCj = 16−jMCj = (w1 + w2 + w3 + w4)6−jMCj, deg 6 homogeneous in w1, w2, w3, w4,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-125
SLIDE 125

Prove MC is non–negative on semi–algebraic set S How? Express MC as a linear comb., with non–negative coeffs., of products of polys wj, non–negative on S. Which wj? w1 = 1 + z, w2 = y − z, w3 = x − y, w4 = −x giving equations of the S facets 1 = w1 + w2 + w3 + w4 (relation) Thus, compute MCj = 16−jMCj = (w1 + w2 + w3 + w4)6−jMCj, deg 6 homogeneous in w1, w2, w3, w4, add from j = 1 to 6

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 17/22

slide-126
SLIDE 126

MC = 192w 5

1w2 + 336w 5 1w3 + 432w 5 1w4 + 768w 4 1w 2 2 + 2112w 4 1w2w3 + 2472w 4 1w2w4 + 1152w 4 1w 2 3 + 2568w 4 1w3w4

+1224w 4

1w 2 4 + 1200w 3 1w 3 2 + 4524w 3 1w 2 2w3 + 5076w 3 1w 2 2w4 + 4824w 3 1w2w 2 3 + 10440w 3 1w2w3w4 + 4992w 3 1w2w 2 4

+1528w 3

1w 3 3 + 4896w 3 1w 2 3w4 + 4740w 3 1w3w 2 4 + 1380w 3 1w 3 4 + 912w 2 1w 4 2 + 4392w 2 1w 3 2w3 + 4830w 2 1w 3 2w4 + 6960w 2 1w 2 2w 2 3

+14850w 2

1w 2 2w3w4 + 7146w 2 1w 2 2w 2 4 + 4442w 2 1w2w 3 3 + 14034w 2 1w2w 2 3w4 + 13656w 2 1w2w3w 2 4 + 4050w 2 1w2w 3 4

+972w 2

1w 4 3 + 4092w 2 1w 3 3w4 + 6072w 2 1w 2 3w 2 4 + 3702w 2 1w3w 3 4 + 774w 2 1w 4 4 + 336w1w 5 2 + 1980w1w 4 2w3 + 2160w1w 4 2w4

+4176w1w 3

2w 2 3 + 8862w1w 3 2w3w4 + 4302w1w 3 2w 2 4 + 4030w1w 2 2w 3 3 + 12657w1w 2 2w 2 3w4 + 12414w1w 2 2w3w 2 4

+3744w1w 2

2w 3 4 + 1790w1w2w 4 3 + 7475w1w2w 3 3w4 + 11163w1w2w 2 3w 2 4 + 6918w1w2w3w 3 4 + 1482w1w2w 4 4 + 292w1w 5 3

+1534w1w 4

3w4 + 3120w1w 3 3w 2 4 + 2988w1w 2 3w 3 4 + 1326w1w3w 4 4 + 216w1w 5 4 + 48w 6 2 + 336w 5 2w3 + 366w 5 2w4 + 888w 4 2w 2 3

+1884w 4

2w3w4 + 924w 4 2w 2 4 + 1152w 3 2w 3 3 + 3615w 3 2w 2 3w4 + 3582w 3 2w3w 2 4 + 1098w 3 2w 3 4 + 776w 2 2w 4 3 + 3233w 2 2w 3 3w4

+4875w 2

2w 2 3w 2 4 + 3072w 2 2w3w 3 4 + 672w 2 2w 4 4 + 256w2w 5 3 + 1340w2w 4 3w4 + 2752w2w 3 3w 2 4 + 2682w2w 2 3w 3 4 + 1218w2w3w 4 4

+204w2w 5

4 + 32w 6 3 + 204w 5 3w4 + 540w 4 3w 2 4 + 728w 3 3w 3 4 + 516w 2 3w 4 4 + 180w3w 5 4 + 24w 6 4.

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 18/22

slide-127
SLIDE 127

MC = 192w 5

1w2 + 336w 5 1w3 + 432w 5 1w4 + 768w 4 1w 2 2 + 2112w 4 1w2w3 + 2472w 4 1w2w4 + 1152w 4 1w 2 3 + 2568w 4 1w3w4

+1224w 4

1w 2 4 + 1200w 3 1w 3 2 + 4524w 3 1w 2 2w3 + 5076w 3 1w 2 2w4 + 4824w 3 1w2w 2 3 + 10440w 3 1w2w3w4 + 4992w 3 1w2w 2 4

+1528w 3

1w 3 3 + 4896w 3 1w 2 3w4 + 4740w 3 1w3w 2 4 + 1380w 3 1w 3 4 + 912w 2 1w 4 2 + 4392w 2 1w 3 2w3 + 4830w 2 1w 3 2w4 + 6960w 2 1w 2 2w 2 3

+14850w 2

1w 2 2w3w4 + 7146w 2 1w 2 2w 2 4 + 4442w 2 1w2w 3 3 + 14034w 2 1w2w 2 3w4 + 13656w 2 1w2w3w 2 4 + 4050w 2 1w2w 3 4

+972w 2

1w 4 3 + 4092w 2 1w 3 3w4 + 6072w 2 1w 2 3w 2 4 + 3702w 2 1w3w 3 4 + 774w 2 1w 4 4 + 336w1w 5 2 + 1980w1w 4 2w3 + 2160w1w 4 2w4

+4176w1w 3

2w 2 3 + 8862w1w 3 2w3w4 + 4302w1w 3 2w 2 4 + 4030w1w 2 2w 3 3 + 12657w1w 2 2w 2 3w4 + 12414w1w 2 2w3w 2 4

+3744w1w 2

2w 3 4 + 1790w1w2w 4 3 + 7475w1w2w 3 3w4 + 11163w1w2w 2 3w 2 4 + 6918w1w2w3w 3 4 + 1482w1w2w 4 4 + 292w1w 5 3

+1534w1w 4

3w4 + 3120w1w 3 3w 2 4 + 2988w1w 2 3w 3 4 + 1326w1w3w 4 4 + 216w1w 5 4 + 48w 6 2 + 336w 5 2w3 + 366w 5 2w4 + 888w 4 2w 2 3

+1884w 4

2w3w4 + 924w 4 2w 2 4 + 1152w 3 2w 3 3 + 3615w 3 2w 2 3w4 + 3582w 3 2w3w 2 4 + 1098w 3 2w 3 4 + 776w 2 2w 4 3 + 3233w 2 2w 3 3w4

+4875w 2

2w 2 3w 2 4 + 3072w 2 2w3w 3 4 + 672w 2 2w 4 4 + 256w2w 5 3 + 1340w2w 4 3w4 + 2752w2w 3 3w 2 4 + 2682w2w 2 3w 3 4 + 1218w2w3w 4 4

+204w2w 5

4 + 32w 6 3 + 204w 5 3w4 + 540w 4 3w 2 4 + 728w 3 3w 3 4 + 516w 2 3w 4 4 + 180w3w 5 4 + 24w 6 4.

All coeffs. are non–negative

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 18/22

slide-128
SLIDE 128

MC = 192w 5

1w2 + 336w 5 1w3 + 432w 5 1w4 + 768w 4 1w 2 2 + 2112w 4 1w2w3 + 2472w 4 1w2w4 + 1152w 4 1w 2 3 + 2568w 4 1w3w4

+1224w 4

1w 2 4 + 1200w 3 1w 3 2 + 4524w 3 1w 2 2w3 + 5076w 3 1w 2 2w4 + 4824w 3 1w2w 2 3 + 10440w 3 1w2w3w4 + 4992w 3 1w2w 2 4

+1528w 3

1w 3 3 + 4896w 3 1w 2 3w4 + 4740w 3 1w3w 2 4 + 1380w 3 1w 3 4 + 912w 2 1w 4 2 + 4392w 2 1w 3 2w3 + 4830w 2 1w 3 2w4 + 6960w 2 1w 2 2w 2 3

+14850w 2

1w 2 2w3w4 + 7146w 2 1w 2 2w 2 4 + 4442w 2 1w2w 3 3 + 14034w 2 1w2w 2 3w4 + 13656w 2 1w2w3w 2 4 + 4050w 2 1w2w 3 4

+972w 2

1w 4 3 + 4092w 2 1w 3 3w4 + 6072w 2 1w 2 3w 2 4 + 3702w 2 1w3w 3 4 + 774w 2 1w 4 4 + 336w1w 5 2 + 1980w1w 4 2w3 + 2160w1w 4 2w4

+4176w1w 3

2w 2 3 + 8862w1w 3 2w3w4 + 4302w1w 3 2w 2 4 + 4030w1w 2 2w 3 3 + 12657w1w 2 2w 2 3w4 + 12414w1w 2 2w3w 2 4

+3744w1w 2

2w 3 4 + 1790w1w2w 4 3 + 7475w1w2w 3 3w4 + 11163w1w2w 2 3w 2 4 + 6918w1w2w3w 3 4 + 1482w1w2w 4 4 + 292w1w 5 3

+1534w1w 4

3w4 + 3120w1w 3 3w 2 4 + 2988w1w 2 3w 3 4 + 1326w1w3w 4 4 + 216w1w 5 4 + 48w 6 2 + 336w 5 2w3 + 366w 5 2w4 + 888w 4 2w 2 3

+1884w 4

2w3w4 + 924w 4 2w 2 4 + 1152w 3 2w 3 3 + 3615w 3 2w 2 3w4 + 3582w 3 2w3w 2 4 + 1098w 3 2w 3 4 + 776w 2 2w 4 3 + 3233w 2 2w 3 3w4

+4875w 2

2w 2 3w 2 4 + 3072w 2 2w3w 3 4 + 672w 2 2w 4 4 + 256w2w 5 3 + 1340w2w 4 3w4 + 2752w2w 3 3w 2 4 + 2682w2w 2 3w 3 4 + 1218w2w3w 4 4

+204w2w 5

4 + 32w 6 3 + 204w 5 3w4 + 540w 4 3w 2 4 + 728w 3 3w 3 4 + 516w 2 3w 4 4 + 180w3w 5 4 + 24w 6 4.

All coeffs. are non–negative (certificate of non–negativeness)

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 18/22

slide-129
SLIDE 129

MC = 192w 5

1w2 + 336w 5 1w3 + 432w 5 1w4 + 768w 4 1w 2 2 + 2112w 4 1w2w3 + 2472w 4 1w2w4 + 1152w 4 1w 2 3 + 2568w 4 1w3w4

+1224w 4

1w 2 4 + 1200w 3 1w 3 2 + 4524w 3 1w 2 2w3 + 5076w 3 1w 2 2w4 + 4824w 3 1w2w 2 3 + 10440w 3 1w2w3w4 + 4992w 3 1w2w 2 4

+1528w 3

1w 3 3 + 4896w 3 1w 2 3w4 + 4740w 3 1w3w 2 4 + 1380w 3 1w 3 4 + 912w 2 1w 4 2 + 4392w 2 1w 3 2w3 + 4830w 2 1w 3 2w4 + 6960w 2 1w 2 2w 2 3

+14850w 2

1w 2 2w3w4 + 7146w 2 1w 2 2w 2 4 + 4442w 2 1w2w 3 3 + 14034w 2 1w2w 2 3w4 + 13656w 2 1w2w3w 2 4 + 4050w 2 1w2w 3 4

+972w 2

1w 4 3 + 4092w 2 1w 3 3w4 + 6072w 2 1w 2 3w 2 4 + 3702w 2 1w3w 3 4 + 774w 2 1w 4 4 + 336w1w 5 2 + 1980w1w 4 2w3 + 2160w1w 4 2w4

+4176w1w 3

2w 2 3 + 8862w1w 3 2w3w4 + 4302w1w 3 2w 2 4 + 4030w1w 2 2w 3 3 + 12657w1w 2 2w 2 3w4 + 12414w1w 2 2w3w 2 4

+3744w1w 2

2w 3 4 + 1790w1w2w 4 3 + 7475w1w2w 3 3w4 + 11163w1w2w 2 3w 2 4 + 6918w1w2w3w 3 4 + 1482w1w2w 4 4 + 292w1w 5 3

+1534w1w 4

3w4 + 3120w1w 3 3w 2 4 + 2988w1w 2 3w 3 4 + 1326w1w3w 4 4 + 216w1w 5 4 + 48w 6 2 + 336w 5 2w3 + 366w 5 2w4 + 888w 4 2w 2 3

+1884w 4

2w3w4 + 924w 4 2w 2 4 + 1152w 3 2w 3 3 + 3615w 3 2w 2 3w4 + 3582w 3 2w3w 2 4 + 1098w 3 2w 3 4 + 776w 2 2w 4 3 + 3233w 2 2w 3 3w4

+4875w 2

2w 2 3w 2 4 + 3072w 2 2w3w 3 4 + 672w 2 2w 4 4 + 256w2w 5 3 + 1340w2w 4 3w4 + 2752w2w 3 3w 2 4 + 2682w2w 2 3w 3 4 + 1218w2w3w 4 4

+204w2w 5

4 + 32w 6 3 + 204w 5 3w4 + 540w 4 3w 2 4 + 728w 3 3w 3 4 + 516w 2 3w 4 4 + 180w3w 5 4 + 24w 6 4.

All coeffs. are non–negative (certificate of non–negativeness) ⇒ Mahler c. holds for alcoved 3–dim.

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 18/22

slide-130
SLIDE 130

MC = 192w 5

1w2 + 336w 5 1w3 + 432w 5 1w4 + 768w 4 1w 2 2 + 2112w 4 1w2w3 + 2472w 4 1w2w4 + 1152w 4 1w 2 3 + 2568w 4 1w3w4

+1224w 4

1w 2 4 + 1200w 3 1w 3 2 + 4524w 3 1w 2 2w3 + 5076w 3 1w 2 2w4 + 4824w 3 1w2w 2 3 + 10440w 3 1w2w3w4 + 4992w 3 1w2w 2 4

+1528w 3

1w 3 3 + 4896w 3 1w 2 3w4 + 4740w 3 1w3w 2 4 + 1380w 3 1w 3 4 + 912w 2 1w 4 2 + 4392w 2 1w 3 2w3 + 4830w 2 1w 3 2w4 + 6960w 2 1w 2 2w 2 3

+14850w 2

1w 2 2w3w4 + 7146w 2 1w 2 2w 2 4 + 4442w 2 1w2w 3 3 + 14034w 2 1w2w 2 3w4 + 13656w 2 1w2w3w 2 4 + 4050w 2 1w2w 3 4

+972w 2

1w 4 3 + 4092w 2 1w 3 3w4 + 6072w 2 1w 2 3w 2 4 + 3702w 2 1w3w 3 4 + 774w 2 1w 4 4 + 336w1w 5 2 + 1980w1w 4 2w3 + 2160w1w 4 2w4

+4176w1w 3

2w 2 3 + 8862w1w 3 2w3w4 + 4302w1w 3 2w 2 4 + 4030w1w 2 2w 3 3 + 12657w1w 2 2w 2 3w4 + 12414w1w 2 2w3w 2 4

+3744w1w 2

2w 3 4 + 1790w1w2w 4 3 + 7475w1w2w 3 3w4 + 11163w1w2w 2 3w 2 4 + 6918w1w2w3w 3 4 + 1482w1w2w 4 4 + 292w1w 5 3

+1534w1w 4

3w4 + 3120w1w 3 3w 2 4 + 2988w1w 2 3w 3 4 + 1326w1w3w 4 4 + 216w1w 5 4 + 48w 6 2 + 336w 5 2w3 + 366w 5 2w4 + 888w 4 2w 2 3

+1884w 4

2w3w4 + 924w 4 2w 2 4 + 1152w 3 2w 3 3 + 3615w 3 2w 2 3w4 + 3582w 3 2w3w 2 4 + 1098w 3 2w 3 4 + 776w 2 2w 4 3 + 3233w 2 2w 3 3w4

+4875w 2

2w 2 3w 2 4 + 3072w 2 2w3w 3 4 + 672w 2 2w 4 4 + 256w2w 5 3 + 1340w2w 4 3w4 + 2752w2w 3 3w 2 4 + 2682w2w 2 3w 3 4 + 1218w2w3w 4 4

+204w2w 5

4 + 32w 6 3 + 204w 5 3w4 + 540w 4 3w 2 4 + 728w 3 3w 3 4 + 516w 2 3w 4 4 + 180w3w 5 4 + 24w 6 4.

All coeffs. are non–negative (certificate of non–negativeness) ⇒ Mahler c. holds for alcoved 3–dim. and equality only attained by boxes

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 18/22

slide-131
SLIDE 131

Summary and future work

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 19/22

slide-132
SLIDE 132

Summary and future work

Volume formula for 3–dim alcoved (rational polynomial)

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 19/22

slide-133
SLIDE 133

Summary and future work

Volume formula for 3–dim alcoved (rational polynomial) Mahler conjecture holds for 3–dim alcoved

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 19/22

slide-134
SLIDE 134

Summary and future work

Volume formula for 3–dim alcoved (rational polynomial) Mahler conjecture holds for 3–dim alcoved Future work

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 19/22

slide-135
SLIDE 135

Summary and future work

Volume formula for 3–dim alcoved (rational polynomial) Mahler conjecture holds for 3–dim alcoved Future work 3–dim alcoved mixed volume formula

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 19/22

slide-136
SLIDE 136

Summary and future work

Volume formula for 3–dim alcoved (rational polynomial) Mahler conjecture holds for 3–dim alcoved Future work 3–dim alcoved mixed volume formula Filling 3–space using alcoved

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 19/22

slide-137
SLIDE 137

Summary and future work

Volume formula for 3–dim alcoved (rational polynomial) Mahler conjecture holds for 3–dim alcoved Future work 3–dim alcoved mixed volume formula Filling 3–space using alcoved In higher dimensions find:

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 19/22

slide-138
SLIDE 138

Summary and future work

Volume formula for 3–dim alcoved (rational polynomial) Mahler conjecture holds for 3–dim alcoved Future work 3–dim alcoved mixed volume formula Filling 3–space using alcoved In higher dimensions find: (a) f –vector,

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 19/22

slide-139
SLIDE 139

Summary and future work

Volume formula for 3–dim alcoved (rational polynomial) Mahler conjecture holds for 3–dim alcoved Future work 3–dim alcoved mixed volume formula Filling 3–space using alcoved In higher dimensions find: (a) f –vector, (b) volume

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 19/22

slide-140
SLIDE 140

Conclusion

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 20/22

slide-141
SLIDE 141

Conclusion

Use NORMAL IDEMPOTENT MATRICES

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 20/22

slide-142
SLIDE 142

Conclusion

Use NORMAL IDEMPOTENT MATRICES (operated TROPICALLY)

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 20/22

slide-143
SLIDE 143

Conclusion

Use NORMAL IDEMPOTENT MATRICES (operated TROPICALLY) to compute and manipulate ALCOVED POLYTOPES

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 20/22

slide-144
SLIDE 144

Conclusion

Use NORMAL IDEMPOTENT MATRICES (operated TROPICALLY) to compute and manipulate ALCOVED POLYTOPES

THANK YOU!

http://www.mat.ucm.es/~mpuente/

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 20/22

slide-145
SLIDE 145

To compute the volume of a polytope is a #P–problem

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 21/22

slide-146
SLIDE 146

To compute the volume of a polytope is a #P–problem (number P or sharp P)

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 21/22

slide-147
SLIDE 147

To compute the volume of a polytope is a #P–problem (number P or sharp P) (Dyer and Frieze 1988, L. Khachiyan 1989)

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 21/22

slide-148
SLIDE 148

To compute the volume of a polytope is a #P–problem (number P or sharp P) (Dyer and Frieze 1988, L. Khachiyan 1989) What is the complexity class #P?

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 21/22

slide-149
SLIDE 149

To compute the volume of a polytope is a #P–problem (number P or sharp P) (Dyer and Frieze 1988, L. Khachiyan 1989) What is the complexity class #P? An NP decision problem:

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 21/22

slide-150
SLIDE 150

To compute the volume of a polytope is a #P–problem (number P or sharp P) (Dyer and Frieze 1988, L. Khachiyan 1989) What is the complexity class #P? An NP decision problem: Are there solutions satisfying...?

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 21/22

slide-151
SLIDE 151

To compute the volume of a polytope is a #P–problem (number P or sharp P) (Dyer and Frieze 1988, L. Khachiyan 1989) What is the complexity class #P? An NP decision problem: Are there solutions satisfying...? The related #P problem:

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 21/22

slide-152
SLIDE 152

To compute the volume of a polytope is a #P–problem (number P or sharp P) (Dyer and Frieze 1988, L. Khachiyan 1989) What is the complexity class #P? An NP decision problem: Are there solutions satisfying...? The related #P problem: How many solutions satisfying... do exist?

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 21/22

slide-153
SLIDE 153

To compute the volume of a polytope is a #P–problem (number P or sharp P) (Dyer and Frieze 1988, L. Khachiyan 1989) What is the complexity class #P? An NP decision problem: Are there solutions satisfying...? The related #P problem: How many solutions satisfying... do exist? Example: In a given list of integers, are there subsets with zero sum?

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 21/22

slide-154
SLIDE 154

To compute the volume of a polytope is a #P–problem (number P or sharp P) (Dyer and Frieze 1988, L. Khachiyan 1989) What is the complexity class #P? An NP decision problem: Are there solutions satisfying...? The related #P problem: How many solutions satisfying... do exist? Example: In a given list of integers, are there subsets with zero sum? is an NP problem

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 21/22

slide-155
SLIDE 155

To compute the volume of a polytope is a #P–problem (number P or sharp P) (Dyer and Frieze 1988, L. Khachiyan 1989) What is the complexity class #P? An NP decision problem: Are there solutions satisfying...? The related #P problem: How many solutions satisfying... do exist? Example: In a given list of integers, are there subsets with zero sum? is an NP problem In a given list of integers, how many subsets with zero sum do exist?

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 21/22

slide-156
SLIDE 156

To compute the volume of a polytope is a #P–problem (number P or sharp P) (Dyer and Frieze 1988, L. Khachiyan 1989) What is the complexity class #P? An NP decision problem: Are there solutions satisfying...? The related #P problem: How many solutions satisfying... do exist? Example: In a given list of integers, are there subsets with zero sum? is an NP problem In a given list of integers, how many subsets with zero sum do exist? the related #P problem

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 21/22

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SLIDE 157

To compute the volume of a polytope is a #P–problem (number P or sharp P) (Dyer and Frieze 1988, L. Khachiyan 1989) What is the complexity class #P? An NP decision problem: Are there solutions satisfying...? The related #P problem: How many solutions satisfying... do exist? Example: In a given list of integers, are there subsets with zero sum? is an NP problem In a given list of integers, how many subsets with zero sum do exist? the related #P problem

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 21/22

slide-158
SLIDE 158

51 terms (depending on x, y, z) 125 terms (depending on wj) Only missing term in MC is w 6

  • 1. Thus the only real root
  • f MC is given by w1 = 1 and w2 = w3 = w4 = 0,

equivalently, by x = y = z = 0. This shows that equality is only attained by boxes, among centrally symmetric alcoved polyhedra. The conjecture also holds for limits of centrally symmetric alcoved polyhedra. f –vector (v, e, f ) ≤ (20, 30, 12) alcoved dodecahedra Permanents of matrix Zonoids

  • Vol. alc. polyhedr. Mahler conj.

M.J. de la Puente, UCM, (Spain) CUNY, July/2018 22/22