real zeros and the alternatingly increasing property in ... · major results and open questions in...
TRANSCRIPT
![Page 1: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/1.jpg)
Real Zeros and the Alternatingly Increasing Property inAlgebraic Combinatorics
Based on joint work with P. Branden (KTH)
Liam Solus
KTH Royal Institute of Technology
6 Augusti 2019NORCOM
Schæffergarden, Kopenhamn, Danmark
![Page 2: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/2.jpg)
Mathematical Aspects of Computer and Information Sciences 201913− 15 November 2019, Gebze-Istanbul, Turkey
http://macis2019.gtu.edu.tr/
Submit papers on combinatorics, designs, graphs, algorithms,codes, cryptography, etc!
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Graphical Models: Conditional Independence and Algebraic Structures23− 25 October 2019, Munich, Germany
https:
//www.groups.ma.tum.de/statistics/allgemeines/veranstaltungen/
graphical-models-conditional-independence-and-algebraic-structures/
![Page 4: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/4.jpg)
Major results and open questions in algebraic and geometric combinatorics centeraround distributional properties of combinatorial generating polynomials:
p := p0 + p1x + · · ·+ pdxd pk ∈ Z≥0, pd 6= 0.
symmetry (w.r.t. m): pk = pm−k .unimodality: p0 ≤ p1 ≤ · · · ≤ pt ≥ · · · ≥ pd−1 ≥ pd .log-concavity: p2
k ≥ pk+1pk−1.γ-positivity: p symmetric with nonnegative coefficients γk in the basis
p =
bd/2c∑k=0
γkxk(x + 1)d−2k .
real-rootedness: p has only real zeros.
For example:(Strong) Mason’s Theorem [Anari, Liu, Gharan, Vinzant, 2018]
[Branden, Huh, 2018]The g -Theorem for Simplicial Spheres [Adiprasito, 2018]Gal’s Conjecture for flag homology spheres [you?, 2019+]
New Property of Interest: Alternatingly Increasing (AI):
0 ≤ p0 ≤ pd ≤ p1 ≤ pd−1 ≤ p2 ≤ · · · ≤ pb d+12 c.
![Page 5: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/5.jpg)
Major results and open questions in algebraic and geometric combinatorics centeraround distributional properties of combinatorial generating polynomials:
p := p0 + p1x + · · ·+ pdxd pk ∈ Z≥0, pd 6= 0.
symmetry (w.r.t. m): pk = pm−k .unimodality: p0 ≤ p1 ≤ · · · ≤ pt ≥ · · · ≥ pd−1 ≥ pd .log-concavity: p2
k ≥ pk+1pk−1.
γ-positivity: p symmetric with nonnegative coefficients γk in the basis
p =
bd/2c∑k=0
γkxk(x + 1)d−2k .
real-rootedness: p has only real zeros.
For example:(Strong) Mason’s Theorem [Anari, Liu, Gharan, Vinzant, 2018]
[Branden, Huh, 2018]The g -Theorem for Simplicial Spheres [Adiprasito, 2018]Gal’s Conjecture for flag homology spheres [you?, 2019+]
New Property of Interest: Alternatingly Increasing (AI):
0 ≤ p0 ≤ pd ≤ p1 ≤ pd−1 ≤ p2 ≤ · · · ≤ pb d+12 c.
![Page 6: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/6.jpg)
Major results and open questions in algebraic and geometric combinatorics centeraround distributional properties of combinatorial generating polynomials:
p := p0 + p1x + · · ·+ pdxd pk ∈ Z≥0, pd 6= 0.
symmetry (w.r.t. m): pk = pm−k .unimodality: p0 ≤ p1 ≤ · · · ≤ pt ≥ · · · ≥ pd−1 ≥ pd .log-concavity: p2
k ≥ pk+1pk−1.γ-positivity: p symmetric with nonnegative coefficients γk in the basis
p =
bd/2c∑k=0
γkxk(x + 1)d−2k .
real-rootedness: p has only real zeros.
For example:(Strong) Mason’s Theorem [Anari, Liu, Gharan, Vinzant, 2018]
[Branden, Huh, 2018]The g -Theorem for Simplicial Spheres [Adiprasito, 2018]Gal’s Conjecture for flag homology spheres [you?, 2019+]
New Property of Interest: Alternatingly Increasing (AI):
0 ≤ p0 ≤ pd ≤ p1 ≤ pd−1 ≤ p2 ≤ · · · ≤ pb d+12 c.
![Page 7: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/7.jpg)
Major results and open questions in algebraic and geometric combinatorics centeraround distributional properties of combinatorial generating polynomials:
p := p0 + p1x + · · ·+ pdxd pk ∈ Z≥0, pd 6= 0.
symmetry (w.r.t. m): pk = pm−k .unimodality: p0 ≤ p1 ≤ · · · ≤ pt ≥ · · · ≥ pd−1 ≥ pd .log-concavity: p2
k ≥ pk+1pk−1.γ-positivity: p symmetric with nonnegative coefficients γk in the basis
p =
bd/2c∑k=0
γkxk(x + 1)d−2k .
real-rootedness: p has only real zeros.
For example:(Strong) Mason’s Theorem [Anari, Liu, Gharan, Vinzant, 2018]
[Branden, Huh, 2018]The g -Theorem for Simplicial Spheres [Adiprasito, 2018]Gal’s Conjecture for flag homology spheres [you?, 2019+]
New Property of Interest: Alternatingly Increasing (AI):
0 ≤ p0 ≤ pd ≤ p1 ≤ pd−1 ≤ p2 ≤ · · · ≤ pb d+12 c.
![Page 8: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/8.jpg)
Major results and open questions in algebraic and geometric combinatorics centeraround distributional properties of combinatorial generating polynomials:
p := p0 + p1x + · · ·+ pdxd pk ∈ Z≥0, pd 6= 0.
symmetry (w.r.t. m): pk = pm−k .unimodality: p0 ≤ p1 ≤ · · · ≤ pt ≥ · · · ≥ pd−1 ≥ pd .log-concavity: p2
k ≥ pk+1pk−1.γ-positivity: p symmetric with nonnegative coefficients γk in the basis
p =
bd/2c∑k=0
γkxk(x + 1)d−2k .
real-rootedness: p has only real zeros.
For example:(Strong) Mason’s Theorem [Anari, Liu, Gharan, Vinzant, 2018]
[Branden, Huh, 2018]
The g -Theorem for Simplicial Spheres [Adiprasito, 2018]Gal’s Conjecture for flag homology spheres [you?, 2019+]
New Property of Interest: Alternatingly Increasing (AI):
0 ≤ p0 ≤ pd ≤ p1 ≤ pd−1 ≤ p2 ≤ · · · ≤ pb d+12 c.
![Page 9: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/9.jpg)
Major results and open questions in algebraic and geometric combinatorics centeraround distributional properties of combinatorial generating polynomials:
p := p0 + p1x + · · ·+ pdxd pk ∈ Z≥0, pd 6= 0.
symmetry (w.r.t. m): pk = pm−k .unimodality: p0 ≤ p1 ≤ · · · ≤ pt ≥ · · · ≥ pd−1 ≥ pd .log-concavity: p2
k ≥ pk+1pk−1.γ-positivity: p symmetric with nonnegative coefficients γk in the basis
p =
bd/2c∑k=0
γkxk(x + 1)d−2k .
real-rootedness: p has only real zeros.
For example:(Strong) Mason’s Theorem [Anari, Liu, Gharan, Vinzant, 2018]
[Branden, Huh, 2018]The g -Theorem for Simplicial Spheres [Adiprasito, 2018]
Gal’s Conjecture for flag homology spheres [you?, 2019+]
New Property of Interest: Alternatingly Increasing (AI):
0 ≤ p0 ≤ pd ≤ p1 ≤ pd−1 ≤ p2 ≤ · · · ≤ pb d+12 c.
![Page 10: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/10.jpg)
Major results and open questions in algebraic and geometric combinatorics centeraround distributional properties of combinatorial generating polynomials:
p := p0 + p1x + · · ·+ pdxd pk ∈ Z≥0, pd 6= 0.
symmetry (w.r.t. m): pk = pm−k .unimodality: p0 ≤ p1 ≤ · · · ≤ pt ≥ · · · ≥ pd−1 ≥ pd .log-concavity: p2
k ≥ pk+1pk−1.γ-positivity: p symmetric with nonnegative coefficients γk in the basis
p =
bd/2c∑k=0
γkxk(x + 1)d−2k .
real-rootedness: p has only real zeros.
For example:(Strong) Mason’s Theorem [Anari, Liu, Gharan, Vinzant, 2018]
[Branden, Huh, 2018]The g -Theorem for Simplicial Spheres [Adiprasito, 2018]Gal’s Conjecture for flag homology spheres [you?, 2019+]
New Property of Interest: Alternatingly Increasing (AI):
0 ≤ p0 ≤ pd ≤ p1 ≤ pd−1 ≤ p2 ≤ · · · ≤ pb d+12 c.
![Page 11: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/11.jpg)
Major results and open questions in algebraic and geometric combinatorics centeraround distributional properties of combinatorial generating polynomials:
p := p0 + p1x + · · ·+ pdxd pk ∈ Z≥0, pd 6= 0.
symmetry (w.r.t. m): pk = pm−k .unimodality: p0 ≤ p1 ≤ · · · ≤ pt ≥ · · · ≥ pd−1 ≥ pd .log-concavity: p2
k ≥ pk+1pk−1.γ-positivity: p symmetric with nonnegative coefficients γk in the basis
p =
bd/2c∑k=0
γkxk(x + 1)d−2k .
real-rootedness: p has only real zeros.
For example:(Strong) Mason’s Theorem [Anari, Liu, Gharan, Vinzant, 2018]
[Branden, Huh, 2018]The g -Theorem for Simplicial Spheres [Adiprasito, 2018]Gal’s Conjecture for flag homology spheres [you?, 2019+]
New Property of Interest: Alternatingly Increasing (AI):
0 ≤ p0 ≤ pd ≤ p1 ≤ pd−1 ≤ p2 ≤ · · · ≤ pb d+12 c.
![Page 12: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/12.jpg)
Symmetric Decompositions:
Lemma. Let p ∈ R[x ] be of degree at most d . Then there exist uniquepolynomials a, b ∈ R[x ] such that
p = a + xb,
deg(a) ≤ d , and a symmetric w.r.t. d ,
deg(b) ≤ d − 1, and b symmetric w.r.t. d − 1
1 + 8x +21x2 +20x3 +5x4 p
1 + 4x + 5x2 + 4x3 + 1x4 a
4x + 16x2 + 16x3 + 4x4 xb
(a, b) is the (symmetric) Id -decomposition of p.
a and b need not have nonnegative coefficients.
p is AI if and only if a and b both unimodal with nonnegative coefficients.
Questions on AI and symmetric decompositions of polynomials are popping up inalgebraic combinatorics!
![Page 13: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/13.jpg)
Symmetric Decompositions:
Lemma. Let p ∈ R[x ] be of degree at most d . Then there exist uniquepolynomials a, b ∈ R[x ] such that
p = a + xb,
deg(a) ≤ d , and a symmetric w.r.t. d ,
deg(b) ≤ d − 1, and b symmetric w.r.t. d − 1
1 + 8x +21x2 +20x3 +5x4 p
1 + 4x + 5x2 + 4x3 + 1x4 a
4x + 16x2 + 16x3 + 4x4 xb
(a, b) is the (symmetric) Id -decomposition of p.
a and b need not have nonnegative coefficients.
p is AI if and only if a and b both unimodal with nonnegative coefficients.
Questions on AI and symmetric decompositions of polynomials are popping up inalgebraic combinatorics!
![Page 14: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/14.jpg)
Symmetric Decompositions:
Lemma. Let p ∈ R[x ] be of degree at most d . Then there exist uniquepolynomials a, b ∈ R[x ] such that
p = a + xb,
deg(a) ≤ d , and a symmetric w.r.t. d ,
deg(b) ≤ d − 1, and b symmetric w.r.t. d − 1
1 + 8x +21x2 +20x3 +5x4 p
1 + 4x + 5x2 + 4x3 + 1x4
a
4x + 16x2 + 16x3 + 4x4
xb
(a, b) is the (symmetric) Id -decomposition of p.
a and b need not have nonnegative coefficients.
p is AI if and only if a and b both unimodal with nonnegative coefficients.
Questions on AI and symmetric decompositions of polynomials are popping up inalgebraic combinatorics!
![Page 15: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/15.jpg)
Symmetric Decompositions:
Lemma. Let p ∈ R[x ] be of degree at most d . Then there exist uniquepolynomials a, b ∈ R[x ] such that
p = a + xb,
deg(a) ≤ d , and a symmetric w.r.t. d ,
deg(b) ≤ d − 1, and b symmetric w.r.t. d − 1
1 + 8x +21x2 +20x3 +5x4 p
1
+ 4x + 5x2 + 4x3 + 1x4
a
4x + 16x2 + 16x3 + 4x4
xb
(a, b) is the (symmetric) Id -decomposition of p.
a and b need not have nonnegative coefficients.
p is AI if and only if a and b both unimodal with nonnegative coefficients.
Questions on AI and symmetric decompositions of polynomials are popping up inalgebraic combinatorics!
![Page 16: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/16.jpg)
Symmetric Decompositions:
Lemma. Let p ∈ R[x ] be of degree at most d . Then there exist uniquepolynomials a, b ∈ R[x ] such that
p = a + xb,
deg(a) ≤ d , and a symmetric w.r.t. d ,
deg(b) ≤ d − 1, and b symmetric w.r.t. d − 1
1 + 8x +21x2 +20x3 +5x4 p
1
+ 4x + 5x2 + 4x3
+ 1x4 a
4x + 16x2 + 16x3 + 4x4
xb
(a, b) is the (symmetric) Id -decomposition of p.
a and b need not have nonnegative coefficients.
p is AI if and only if a and b both unimodal with nonnegative coefficients.
Questions on AI and symmetric decompositions of polynomials are popping up inalgebraic combinatorics!
![Page 17: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/17.jpg)
Symmetric Decompositions:
Lemma. Let p ∈ R[x ] be of degree at most d . Then there exist uniquepolynomials a, b ∈ R[x ] such that
p = a + xb,
deg(a) ≤ d , and a symmetric w.r.t. d ,
deg(b) ≤ d − 1, and b symmetric w.r.t. d − 1
1 + 8x +21x2 +20x3 +5x4 p
1
+ 4x + 5x2 + 4x3
+ 1x4 a
4x + 16x2 + 16x3
+ 4x4 xb
(a, b) is the (symmetric) Id -decomposition of p.
a and b need not have nonnegative coefficients.
p is AI if and only if a and b both unimodal with nonnegative coefficients.
Questions on AI and symmetric decompositions of polynomials are popping up inalgebraic combinatorics!
![Page 18: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/18.jpg)
Symmetric Decompositions:
Lemma. Let p ∈ R[x ] be of degree at most d . Then there exist uniquepolynomials a, b ∈ R[x ] such that
p = a + xb,
deg(a) ≤ d , and a symmetric w.r.t. d ,
deg(b) ≤ d − 1, and b symmetric w.r.t. d − 1
1 + 8x +21x2 +20x3 +5x4 p
1
+ 4x + 5x2 + 4x3
+ 1x4 a
4x
+ 16x2 + 16x3
+ 4x4 xb
(a, b) is the (symmetric) Id -decomposition of p.
a and b need not have nonnegative coefficients.
p is AI if and only if a and b both unimodal with nonnegative coefficients.
Questions on AI and symmetric decompositions of polynomials are popping up inalgebraic combinatorics!
![Page 19: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/19.jpg)
Symmetric Decompositions:
Lemma. Let p ∈ R[x ] be of degree at most d . Then there exist uniquepolynomials a, b ∈ R[x ] such that
p = a + xb,
deg(a) ≤ d , and a symmetric w.r.t. d ,
deg(b) ≤ d − 1, and b symmetric w.r.t. d − 1
1 + 8x +21x2 +20x3 +5x4 p
1 + 4x
+ 5x2 + 4x3
+ 1x4 a
4x
+ 16x2 + 16x3
+ 4x4 xb
(a, b) is the (symmetric) Id -decomposition of p.
a and b need not have nonnegative coefficients.
p is AI if and only if a and b both unimodal with nonnegative coefficients.
Questions on AI and symmetric decompositions of polynomials are popping up inalgebraic combinatorics!
![Page 20: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/20.jpg)
Symmetric Decompositions:
Lemma. Let p ∈ R[x ] be of degree at most d . Then there exist uniquepolynomials a, b ∈ R[x ] such that
p = a + xb,
deg(a) ≤ d , and a symmetric w.r.t. d ,
deg(b) ≤ d − 1, and b symmetric w.r.t. d − 1
1 + 8x +21x2 +20x3 +5x4 p
1 + 4x
+ 5x2
+ 4x3 + 1x4 a
4x
+ 16x2 + 16x3
+ 4x4 xb
(a, b) is the (symmetric) Id -decomposition of p.
a and b need not have nonnegative coefficients.
p is AI if and only if a and b both unimodal with nonnegative coefficients.
Questions on AI and symmetric decompositions of polynomials are popping up inalgebraic combinatorics!
![Page 21: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/21.jpg)
Symmetric Decompositions:
Lemma. Let p ∈ R[x ] be of degree at most d . Then there exist uniquepolynomials a, b ∈ R[x ] such that
p = a + xb,
deg(a) ≤ d , and a symmetric w.r.t. d ,
deg(b) ≤ d − 1, and b symmetric w.r.t. d − 1
1 + 8x +21x2 +20x3 +5x4 p
1 + 4x
+ 5x2
+ 4x3 + 1x4 a
4x
+ 16x2
+ 16x3 + 4x4 xb
(a, b) is the (symmetric) Id -decomposition of p.
a and b need not have nonnegative coefficients.
p is AI if and only if a and b both unimodal with nonnegative coefficients.
Questions on AI and symmetric decompositions of polynomials are popping up inalgebraic combinatorics!
![Page 22: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/22.jpg)
Symmetric Decompositions:
Lemma. Let p ∈ R[x ] be of degree at most d . Then there exist uniquepolynomials a, b ∈ R[x ] such that
p = a + xb,
deg(a) ≤ d , and a symmetric w.r.t. d ,
deg(b) ≤ d − 1, and b symmetric w.r.t. d − 1
1 + 8x +21x2 +20x3 +5x4 p
1 + 4x
+ 5x2
+ 4x3 + 1x4 a
4x + 16x2 + 16x3 + 4x4 xb
(a, b) is the (symmetric) Id -decomposition of p.
a and b need not have nonnegative coefficients.
p is AI if and only if a and b both unimodal with nonnegative coefficients.
Questions on AI and symmetric decompositions of polynomials are popping up inalgebraic combinatorics!
![Page 23: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/23.jpg)
Symmetric Decompositions:
Lemma. Let p ∈ R[x ] be of degree at most d . Then there exist uniquepolynomials a, b ∈ R[x ] such that
p = a + xb,
deg(a) ≤ d , and a symmetric w.r.t. d ,
deg(b) ≤ d − 1, and b symmetric w.r.t. d − 1
1 + 8x +21x2 +20x3 +5x4 p
1 + 4x + 5x2 + 4x3 + 1x4 a
4x + 16x2 + 16x3 + 4x4 xb
(a, b) is the (symmetric) Id -decomposition of p.
a and b need not have nonnegative coefficients.
p is AI if and only if a and b both unimodal with nonnegative coefficients.
Questions on AI and symmetric decompositions of polynomials are popping up inalgebraic combinatorics!
![Page 24: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/24.jpg)
Symmetric Decompositions:
Lemma. Let p ∈ R[x ] be of degree at most d . Then there exist uniquepolynomials a, b ∈ R[x ] such that
p = a + xb,
deg(a) ≤ d , and a symmetric w.r.t. d ,
deg(b) ≤ d − 1, and b symmetric w.r.t. d − 1
1 + 8x +21x2 +20x3 +5x4 p
1 + 4x + 5x2 + 4x3 + 1x4 a
4x + 16x2 + 16x3 + 4x4 xb
(a, b) is the (symmetric) Id -decomposition of p.
a and b need not have nonnegative coefficients.
p is AI if and only if a and b both unimodal with nonnegative coefficients.
Questions on AI and symmetric decompositions of polynomials are popping up inalgebraic combinatorics!
![Page 25: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/25.jpg)
Symmetric Decompositions:
Lemma. Let p ∈ R[x ] be of degree at most d . Then there exist uniquepolynomials a, b ∈ R[x ] such that
p = a + xb,
deg(a) ≤ d , and a symmetric w.r.t. d ,
deg(b) ≤ d − 1, and b symmetric w.r.t. d − 1
1 + 8x +21x2 +20x3 +5x4 p
1 + 4x + 5x2 + 4x3 + 1x4 a
4x + 16x2 + 16x3 + 4x4 xb
(a, b) is the (symmetric) Id -decomposition of p.
a and b need not have nonnegative coefficients.
p is AI if and only if a and b both unimodal with nonnegative coefficients.
Questions on AI and symmetric decompositions of polynomials are popping up inalgebraic combinatorics!
![Page 26: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/26.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 27: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/27.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 28: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/28.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 29: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/29.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 30: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/30.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 31: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/31.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 32: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/32.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 33: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/33.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 34: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/34.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 35: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/35.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 36: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/36.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 37: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/37.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 38: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/38.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 39: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/39.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 40: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/40.jpg)
Subdivisions of Simplices:
Let Ω be a d-dimensional simplicial complex that subdivides the d-simplex ∆d .
barycentric
subdivision
3rd edgewise
subdivision
`d(Ω; x) :=∑F⊂∆d
(−1)d−dim(F )h(Ω|F ; x)
is the local h-polynomial of Ω.
`d(Ω; x) is (often) symmetric w.r.t. d + 1 with nonnegative coefficients.
`d(Ω; x) is unimodal when Ω is a regular subdivision.
When is `d(Ω : x) further γ-positive?
Ωn,r := the r th edgewise subdivision of the barycentric subdivision of ∆n.
Question (Athanasiadis). Is `n(Ωn,r ; x) γ-positive?
![Page 41: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/41.jpg)
Subdivisions of Simplices:
Answer: Yes, via a combinatorial proof:
dn,r :=∑
(π,c)∈Dn,r
xexc(π,c),
the nth colored derangement polynomial:
Dn,r is all colored derangements:
(π, c) ∼ πc11 π
c22 · · ·π
cnn ,
where ck ∈ 0, 1, . . . , r − 1, and πi 6= i if ci = 0.
exc(π, c) := |k ∈ [n] : πk ≥ k| .
Theorem (Athanasiadis, 2014). If (a, b) is the symmetric In-decomposition ofdn,r then a and b are γ-positive and a = `n(Ωn,r ; x).
Note: If p is symmetric and real-rooted then p is γ-positive.
Conjecture (Athanasiadis; 2017). a and b are also real-rooted.
![Page 42: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/42.jpg)
Subdivisions of Simplices:
Answer: Yes, via a combinatorial proof:
dn,r :=∑
(π,c)∈Dn,r
xexc(π,c),
the nth colored derangement polynomial:
Dn,r is all colored derangements:
(π, c) ∼ πc11 π
c22 · · ·π
cnn ,
where ck ∈ 0, 1, . . . , r − 1, and πi 6= i if ci = 0.
exc(π, c) := |k ∈ [n] : πk ≥ k| .
Theorem (Athanasiadis, 2014). If (a, b) is the symmetric In-decomposition ofdn,r then a and b are γ-positive and a = `n(Ωn,r ; x).
Note: If p is symmetric and real-rooted then p is γ-positive.
Conjecture (Athanasiadis; 2017). a and b are also real-rooted.
![Page 43: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/43.jpg)
Subdivisions of Simplices:
Answer: Yes, via a combinatorial proof:
dn,r :=∑
(π,c)∈Dn,r
xexc(π,c),
the nth colored derangement polynomial:
Dn,r is all colored derangements:
(π, c) ∼ πc11 π
c22 · · ·π
cnn ,
where ck ∈ 0, 1, . . . , r − 1, and πi 6= i if ci = 0.
exc(π, c) := |k ∈ [n] : πk ≥ k| .
Theorem (Athanasiadis, 2014). If (a, b) is the symmetric In-decomposition ofdn,r then a and b are γ-positive and a = `n(Ωn,r ; x).
Note: If p is symmetric and real-rooted then p is γ-positive.
Conjecture (Athanasiadis; 2017). a and b are also real-rooted.
![Page 44: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/44.jpg)
Subdivisions of Simplices:
Answer: Yes, via a combinatorial proof:
dn,r :=∑
(π,c)∈Dn,r
xexc(π,c),
the nth colored derangement polynomial:
Dn,r is all colored derangements:
(π, c) ∼ πc11 π
c22 · · ·π
cnn ,
where ck ∈ 0, 1, . . . , r − 1, and πi 6= i if ci = 0.
exc(π, c) := |k ∈ [n] : πk ≥ k| .
Theorem (Athanasiadis, 2014). If (a, b) is the symmetric In-decomposition ofdn,r then a and b are γ-positive and a = `n(Ωn,r ; x).
Note: If p is symmetric and real-rooted then p is γ-positive.
Conjecture (Athanasiadis; 2017). a and b are also real-rooted.
![Page 45: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/45.jpg)
Subdivisions of Simplices:
Answer: Yes, via a combinatorial proof:
dn,r :=∑
(π,c)∈Dn,r
xexc(π,c),
the nth colored derangement polynomial:
Dn,r is all colored derangements:
(π, c) ∼ πc11 π
c22 · · ·π
cnn ,
where ck ∈ 0, 1, . . . , r − 1, and πi 6= i if ci = 0.
exc(π, c) := |k ∈ [n] : πk ≥ k| .
Theorem (Athanasiadis, 2014). If (a, b) is the symmetric In-decomposition ofdn,r then a and b are γ-positive and a = `n(Ωn,r ; x).
Note: If p is symmetric and real-rooted then p is γ-positive.
Conjecture (Athanasiadis; 2017). a and b are also real-rooted.
![Page 46: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/46.jpg)
Subdivisions of Simplices:
Answer: Yes, via a combinatorial proof:
dn,r :=∑
(π,c)∈Dn,r
xexc(π,c),
the nth colored derangement polynomial:
Dn,r is all colored derangements:
(π, c) ∼ πc11 π
c22 · · ·π
cnn ,
where ck ∈ 0, 1, . . . , r − 1, and πi 6= i if ci = 0.
exc(π, c) := |k ∈ [n] : πk ≥ k| .
Theorem (Athanasiadis, 2014). If (a, b) is the symmetric In-decomposition ofdn,r then a and b are γ-positive and a = `n(Ωn,r ; x).
Note: If p is symmetric and real-rooted then p is γ-positive.
Conjecture (Athanasiadis; 2017). a and b are also real-rooted.
![Page 47: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/47.jpg)
Subdivisions of Simplices:
Answer: Yes, via a combinatorial proof:
dn,r :=∑
(π,c)∈Dn,r
xexc(π,c),
the nth colored derangement polynomial:
Dn,r is all colored derangements:
(π, c) ∼ πc11 π
c22 · · ·π
cnn ,
where ck ∈ 0, 1, . . . , r − 1, and πi 6= i if ci = 0.
exc(π, c) := |k ∈ [n] : πk ≥ k| .
Theorem (Athanasiadis, 2014). If (a, b) is the symmetric In-decomposition ofdn,r then a and b are γ-positive and a = `n(Ωn,r ; x).
Note: If p is symmetric and real-rooted then p is γ-positive.
Conjecture (Athanasiadis; 2017). a and b are also real-rooted.
![Page 48: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/48.jpg)
Ehrhart Theory:
P ⊂ Rn a d-dimensional lattice polytope.
kP := kp ∈ Rn : p ∈ P.
Ak := spanCxα1
1 · · · xαnn tk ∈ C[x±1 , . . . , x
±n , t] : (α1, . . . , αn) ∈ kP ∩ Zn
The Ehrhart ring of P:
AP :=∞⊕k=0
Ak A0 := C.
If we consider the Hilbert series
1 +∑k>0
dimC(Ak)xk =h∗(P; x)
(1− x)d+1,
then h∗(P; x) is called the h∗-polynomial of P.
Conjecture (Stanley, 1989; Hibi and Ohsugi, 2001; etc...). If AP is integrallyclosed then h∗(P; x) is unimodal.
![Page 49: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/49.jpg)
Ehrhart Theory:
P ⊂ Rn a d-dimensional lattice polytope.
kP := kp ∈ Rn : p ∈ P.
Ak := spanCxα1
1 · · · xαnn tk ∈ C[x±1 , . . . , x
±n , t] : (α1, . . . , αn) ∈ kP ∩ Zn
The Ehrhart ring of P:
AP :=∞⊕k=0
Ak A0 := C.
If we consider the Hilbert series
1 +∑k>0
dimC(Ak)xk =h∗(P; x)
(1− x)d+1,
then h∗(P; x) is called the h∗-polynomial of P.
Conjecture (Stanley, 1989; Hibi and Ohsugi, 2001; etc...). If AP is integrallyclosed then h∗(P; x) is unimodal.
![Page 50: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/50.jpg)
Ehrhart Theory:
P ⊂ Rn a d-dimensional lattice polytope.
kP := kp ∈ Rn : p ∈ P.
Ak := spanCxα1
1 · · · xαnn tk ∈ C[x±1 , . . . , x
±n , t] : (α1, . . . , αn) ∈ kP ∩ Zn
The Ehrhart ring of P:
AP :=∞⊕k=0
Ak A0 := C.
If we consider the Hilbert series
1 +∑k>0
dimC(Ak)xk =h∗(P; x)
(1− x)d+1,
then h∗(P; x) is called the h∗-polynomial of P.
Conjecture (Stanley, 1989; Hibi and Ohsugi, 2001; etc...). If AP is integrallyclosed then h∗(P; x) is unimodal.
![Page 51: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/51.jpg)
Ehrhart Theory:
P ⊂ Rn a d-dimensional lattice polytope.
kP := kp ∈ Rn : p ∈ P.
Ak := spanCxα1
1 · · · xαnn tk ∈ C[x±1 , . . . , x
±n , t] : (α1, . . . , αn) ∈ kP ∩ Zn
The Ehrhart ring of P:
AP :=∞⊕k=0
Ak A0 := C.
If we consider the Hilbert series
1 +∑k>0
dimC(Ak)xk =h∗(P; x)
(1− x)d+1,
then h∗(P; x) is called the h∗-polynomial of P.
Conjecture (Stanley, 1989; Hibi and Ohsugi, 2001; etc...). If AP is integrallyclosed then h∗(P; x) is unimodal.
![Page 52: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/52.jpg)
Ehrhart Theory:
P ⊂ Rn a d-dimensional lattice polytope.
kP := kp ∈ Rn : p ∈ P.
Ak := spanCxα1
1 · · · xαnn tk ∈ C[x±1 , . . . , x
±n , t] : (α1, . . . , αn) ∈ kP ∩ Zn
The Ehrhart ring of P:
AP :=∞⊕k=0
Ak A0 := C.
If we consider the Hilbert series
1 +∑k>0
dimC(Ak)xk =h∗(P; x)
(1− x)d+1,
then h∗(P; x) is called the h∗-polynomial of P.
Conjecture (Stanley, 1989; Hibi and Ohsugi, 2001; etc...). If AP is integrallyclosed then h∗(P; x) is unimodal.
![Page 53: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/53.jpg)
Ehrhart Theory:
P ⊂ Rn a d-dimensional lattice polytope.
kP := kp ∈ Rn : p ∈ P.
Ak := spanCxα1
1 · · · xαnn tk ∈ C[x±1 , . . . , x
±n , t] : (α1, . . . , αn) ∈ kP ∩ Zn
The Ehrhart ring of P:
AP :=∞⊕k=0
Ak A0 := C.
If we consider the Hilbert series
1 +∑k>0
dimC(Ak)xk =h∗(P; x)
(1− x)d+1,
then h∗(P; x) is called the h∗-polynomial of P.
Conjecture (Stanley, 1989; Hibi and Ohsugi, 2001; etc...). If AP is integrallyclosed then h∗(P; x) is unimodal.
![Page 54: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/54.jpg)
Ehrhart Theory:
Theorem (Schepers and Van Langenhoven, 2013). h∗(P; x) is unimodalwhenever P is a parallelepiped. Moreover, if P contains interior lattice pointsthen h∗(P; x) is AI.
Question (Schepers and Van Langenhoven, 2013). If AP is integrally closedand P contains interior lattice points then is h∗(P; x) AI?
(Beck, Jochemko, McCullough, 2016). Yes, for centrally symmetric latticezonotopes.
Conjecture (Beck et al., 2016). True for all lattice zonotopes?
![Page 55: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/55.jpg)
Ehrhart Theory:
Theorem (Schepers and Van Langenhoven, 2013). h∗(P; x) is unimodalwhenever P is a parallelepiped. Moreover, if P contains interior lattice pointsthen h∗(P; x) is AI.
Question (Schepers and Van Langenhoven, 2013). If AP is integrally closedand P contains interior lattice points then is h∗(P; x) AI?
(Beck, Jochemko, McCullough, 2016). Yes, for centrally symmetric latticezonotopes.
Conjecture (Beck et al., 2016). True for all lattice zonotopes?
![Page 56: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/56.jpg)
Ehrhart Theory:
Theorem (Schepers and Van Langenhoven, 2013). h∗(P; x) is unimodalwhenever P is a parallelepiped. Moreover, if P contains interior lattice pointsthen h∗(P; x) is AI.
Question (Schepers and Van Langenhoven, 2013). If AP is integrally closedand P contains interior lattice points then is h∗(P; x) AI?
(Beck, Jochemko, McCullough, 2016). Yes, for centrally symmetric latticezonotopes.
Conjecture (Beck et al., 2016). True for all lattice zonotopes?
![Page 57: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/57.jpg)
Ehrhart Theory:
Theorem (Schepers and Van Langenhoven, 2013). h∗(P; x) is unimodalwhenever P is a parallelepiped. Moreover, if P contains interior lattice pointsthen h∗(P; x) is AI.
Question (Schepers and Van Langenhoven, 2013). If AP is integrally closedand P contains interior lattice points then is h∗(P; x) AI?
(Beck, Jochemko, McCullough, 2016). Yes, for centrally symmetric latticezonotopes.
Conjecture (Beck et al., 2016). True for all lattice zonotopes?
![Page 58: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/58.jpg)
Ehrhart Theory:
Theorem (Schepers and Van Langenhoven, 2013). h∗(P; x) is unimodalwhenever P is a parallelepiped. Moreover, if P contains interior lattice pointsthen h∗(P; x) is AI.
Question (Schepers and Van Langenhoven, 2013). If AP is integrally closedand P contains interior lattice points then is h∗(P; x) AI?
(Beck, Jochemko, McCullough, 2016). Yes, for centrally symmetric latticezonotopes.
Conjecture (Beck et al., 2016). True for all lattice zonotopes?
![Page 59: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/59.jpg)
Real Zeros and Two Theorems on AI:
p, q ∈ R[x ] with only real zeros
α1 ≥ α2 ≥ . . . ≥ αd and β1 ≥ β2 ≥ . . . ≥ βm,
respectively.
q ≺ p if α1 ≥ β1 ≥ α2 ≥ β2 · · · .
Note: To show p is AI we can show b ≺ a.
Theorem (Branden, LS; 2019). If p has Id -decomposition (a, b) such that botha and b have only nonnegative coefficients then the following are equivalent:
1 b ≺ a
2 a ≺ p
3 b ≺ p
4 xdp(1/x) ≺ p
Corollary. b ≺ a for (a, b) the Id -decomposition of dn,r .
![Page 60: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/60.jpg)
Real Zeros and Two Theorems on AI:
p, q ∈ R[x ] with only real zeros
α1 ≥ α2 ≥ . . . ≥ αd and β1 ≥ β2 ≥ . . . ≥ βm,
respectively.
q ≺ p if α1 ≥ β1 ≥ α2 ≥ β2 · · · .
Note: To show p is AI we can show b ≺ a.
Theorem (Branden, LS; 2019). If p has Id -decomposition (a, b) such that botha and b have only nonnegative coefficients then the following are equivalent:
1 b ≺ a
2 a ≺ p
3 b ≺ p
4 xdp(1/x) ≺ p
Corollary. b ≺ a for (a, b) the Id -decomposition of dn,r .
![Page 61: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/61.jpg)
Real Zeros and Two Theorems on AI:
p, q ∈ R[x ] with only real zeros
α1 ≥ α2 ≥ . . . ≥ αd and β1 ≥ β2 ≥ . . . ≥ βm,
respectively.
q ≺ p if α1 ≥ β1 ≥ α2 ≥ β2 · · · .
Note: To show p is AI we can show b ≺ a.
Theorem (Branden, LS; 2019). If p has Id -decomposition (a, b) such that botha and b have only nonnegative coefficients then the following are equivalent:
1 b ≺ a
2 a ≺ p
3 b ≺ p
4 xdp(1/x) ≺ p
Corollary. b ≺ a for (a, b) the Id -decomposition of dn,r .
![Page 62: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/62.jpg)
Real Zeros and Two Theorems on AI:
p, q ∈ R[x ] with only real zeros
α1 ≥ α2 ≥ . . . ≥ αd and β1 ≥ β2 ≥ . . . ≥ βm,
respectively.
q ≺ p if α1 ≥ β1 ≥ α2 ≥ β2 · · · .
Note: To show p is AI we can show b ≺ a.
Theorem (Branden, LS; 2019). If p has Id -decomposition (a, b) such that botha and b have only nonnegative coefficients then the following are equivalent:
1 b ≺ a
2 a ≺ p
3 b ≺ p
4 xdp(1/x) ≺ p
Corollary. b ≺ a for (a, b) the Id -decomposition of dn,r .
![Page 63: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/63.jpg)
Real Zeros and Two Theorems on AI:
p, q ∈ R[x ] with only real zeros
α1 ≥ α2 ≥ . . . ≥ αd and β1 ≥ β2 ≥ . . . ≥ βm,
respectively.
q ≺ p if α1 ≥ β1 ≥ α2 ≥ β2 · · · .
Note: To show p is AI we can show b ≺ a.
Theorem (Branden, LS; 2019). If p has Id -decomposition (a, b) such that botha and b have only nonnegative coefficients then the following are equivalent:
1 b ≺ a
2 a ≺ p
3 b ≺ p
4 xdp(1/x) ≺ p
Corollary. b ≺ a for (a, b) the Id -decomposition of dn,r .
![Page 64: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/64.jpg)
Real Zeros and Two Theorems on AI:
p, q ∈ R[x ] with only real zeros
α1 ≥ α2 ≥ . . . ≥ αd and β1 ≥ β2 ≥ . . . ≥ βm,
respectively.
q ≺ p if α1 ≥ β1 ≥ α2 ≥ β2 · · · .
Note: To show p is AI we can show b ≺ a.
Theorem (Branden, LS; 2019). If p has Id -decomposition (a, b) such that botha and b have only nonnegative coefficients then the following are equivalent:
1 b ≺ a
2 a ≺ p
3 b ≺ p
4 xdp(1/x) ≺ p
Corollary. b ≺ a for (a, b) the Id -decomposition of dn,r .
![Page 65: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/65.jpg)
Real Zeros and Two Theorems on AI:
p, q ∈ R[x ] with only real zeros
α1 ≥ α2 ≥ . . . ≥ αd and β1 ≥ β2 ≥ . . . ≥ βm,
respectively.
q ≺ p if α1 ≥ β1 ≥ α2 ≥ β2 · · · .
Note: To show p is AI we can show b ≺ a.
Theorem (Branden, LS; 2019). If p has Id -decomposition (a, b) such that botha and b have only nonnegative coefficients then the following are equivalent:
1 b ≺ a
2 a ≺ p
3 b ≺ p
4 xdp(1/x) ≺ p
Corollary. b ≺ a for (a, b) the Id -decomposition of dn,r .
![Page 66: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/66.jpg)
Real Zeros and Two Theorems on AI:
Theorem (Branden, LS; 2019). Let
i =d∑
k=0
ckxk(x + 1)d−k
with ck ≥ 0 and let
1 +∑m>0
i(m)xm =h(x)
(1− x)d+1.
Ifc0 + · · ·+ cj ≤ cd + · · ·+ cd−j
for 0 ≤ j ≤ d/2, then both a and b in the Id -decomposition (a, b) of h arereal-rooted.
(Dall; 2015) showed that when h is the h∗-polynomial of a lattice zonotope, thecoefficients ck are the coefficients of the h∗-polynomial of a Lawrence polytope.
Corollary. If P is a lattice zonotope with interior lattice points then h∗(P; x) isAI. In fact both a and b in the Id -decomposition of h∗(P; x) are real-rooted.
![Page 67: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/67.jpg)
Real Zeros and Two Theorems on AI:
Theorem (Branden, LS; 2019). Let
i =d∑
k=0
ckxk(x + 1)d−k
with ck ≥ 0 and let
1 +∑m>0
i(m)xm =h(x)
(1− x)d+1.
Ifc0 + · · ·+ cj ≤ cd + · · ·+ cd−j
for 0 ≤ j ≤ d/2, then both a and b in the Id -decomposition (a, b) of h arereal-rooted.
(Dall; 2015) showed that when h is the h∗-polynomial of a lattice zonotope, thecoefficients ck are the coefficients of the h∗-polynomial of a Lawrence polytope.
Corollary. If P is a lattice zonotope with interior lattice points then h∗(P; x) isAI. In fact both a and b in the Id -decomposition of h∗(P; x) are real-rooted.
![Page 68: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/68.jpg)
Real Zeros and Two Theorems on AI:
Theorem (Branden, LS; 2019). Let
i =d∑
k=0
ckxk(x + 1)d−k
with ck ≥ 0 and let
1 +∑m>0
i(m)xm =h(x)
(1− x)d+1.
Ifc0 + · · ·+ cj ≤ cd + · · ·+ cd−j
for 0 ≤ j ≤ d/2, then both a and b in the Id -decomposition (a, b) of h arereal-rooted.
(Dall; 2015) showed that when h is the h∗-polynomial of a lattice zonotope, thecoefficients ck are the coefficients of the h∗-polynomial of a Lawrence polytope.
Corollary. If P is a lattice zonotope with interior lattice points then h∗(P; x) isAI. In fact both a and b in the Id -decomposition of h∗(P; x) are real-rooted.
![Page 69: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/69.jpg)
Real Zeros and Two Theorems on AI:
Theorem (Branden, LS; 2019). Let
i =d∑
k=0
ckxk(x + 1)d−k
with ck ≥ 0 and let
1 +∑m>0
i(m)xm =h(x)
(1− x)d+1.
Ifc0 + · · ·+ cj ≤ cd + · · ·+ cd−j
for 0 ≤ j ≤ d/2, then both a and b in the Id -decomposition (a, b) of h arereal-rooted.
(Dall; 2015) showed that when h is the h∗-polynomial of a lattice zonotope, thecoefficients ck are the coefficients of the h∗-polynomial of a Lawrence polytope.
Corollary. If P is a lattice zonotope with interior lattice points then h∗(P; x) isAI. In fact both a and b in the Id -decomposition of h∗(P; x) are real-rooted.
![Page 70: Real Zeros and the Alternatingly Increasing Property in ... · Major results and open questions in algebraic and geometric combinatorics center around distributional properties of](https://reader034.vdocuments.mx/reader034/viewer/2022050113/5f49fea791bb81620f67128b/html5/thumbnails/70.jpg)
Thank you for listening!
Please check out our paper:
P. Branden and L. Solus. Symmetric decompositions and real-rootedness. To appear in InternationalMathematics Research Notices IMRN (2019).