Ehrhart Positivity Federico Castillo University of California, Davis Joint work with Fu Liu December 15, 2016 Federico Castillo UC Davis Ehrhart Positivity
Lattice points of a polytope A (convex) polytope is a bounded solution set of a finite system of linear inequalities, or is the convex hull of a finite set of points. Federico Castillo UC Davis Ehrhart Positivity
Lattice points of a polytope A (convex) polytope is a bounded solution set of a finite system of linear inequalities, or is the convex hull of a finite set of points. An integral polytope is a polytope whose vertices are all lattice points. i.e., points with integer coordinates. Federico Castillo UC Davis Ehrhart Positivity
Lattice points of a polytope A (convex) polytope is a bounded solution set of a finite system of linear inequalities, or is the convex hull of a finite set of points. An integral polytope is a polytope whose vertices are all lattice points. i.e., points with integer coordinates. Definition For any polytope P ⊂ R d and positive integer m ∈ N , the m th dilation of P is mP = { mx : x ∈ P } . We define i ( P , m ) = | mP ∩ Z d | to be the number of lattice points in the mP . Federico Castillo UC Davis Ehrhart Positivity
Example P 3P Federico Castillo UC Davis Ehrhart Positivity
Example P 3P In this example we can see that i ( P , m ) = ( m + 1 ) 2 Federico Castillo UC Davis Ehrhart Positivity
Theorem of Ehrhart (on integral polytopes) Figure: Eugene Ehrhart. Federico Castillo UC Davis Ehrhart Positivity
Theorem of Ehrhart (on integral polytopes) Figure: Eugene Ehrhart. Theorem[Ehrhart] Let P be a d -dimensional integral polytope. Then i ( P , m ) is a polynomial in m of degree d . Federico Castillo UC Davis Ehrhart Positivity
Theorem of Ehrhart (on integral polytopes) Figure: Eugene Ehrhart. Theorem[Ehrhart] Let P be a d -dimensional integral polytope. Then i ( P , m ) is a polynomial in m of degree d . Federico Castillo UC Davis Ehrhart Positivity
The h ∗ or δ vector. Therefore, we call i ( P , m ) the Ehrhart polynomial of P . Federico Castillo UC Davis Ehrhart Positivity
The h ∗ or δ vector. Therefore, we call i ( P , m ) the Ehrhart polynomial of P . We study its coefficients. Federico Castillo UC Davis Ehrhart Positivity
The h ∗ or δ vector. Therefore, we call i ( P , m ) the Ehrhart polynomial of P . We study its coefficients. ... however, there is another popular point of view. Federico Castillo UC Davis Ehrhart Positivity
The h ∗ or δ vector. Therefore, we call i ( P , m ) the Ehrhart polynomial of P . We study its coefficients. ... however, there is another popular point of view. The fact that i ( P , m ) is a polynomial with integer values at integer points suggests other forms of expanding it. An alternative basis We can write: � m + d � � m + d − 1 � � m � i ( P , m ) = h ∗ + h ∗ + · · · + h ∗ 0 ( P ) 1 ( P ) d ( P ) . d d d Federico Castillo UC Davis Ehrhart Positivity
More on the the h ∗ or δ vector. The vector ( h ∗ 0 , h ∗ 1 , · · · , h ∗ d ) has many good properties. Theorem(Stanley) For any lattice polytope P , h ∗ i ( P ) is nonnegative integer. Federico Castillo UC Davis Ehrhart Positivity
More on the the h ∗ or δ vector. The vector ( h ∗ 0 , h ∗ 1 , · · · , h ∗ d ) has many good properties. Theorem(Stanley) For any lattice polytope P , h ∗ i ( P ) is nonnegative integer. Additionally it has an algebraic meaning. Federico Castillo UC Davis Ehrhart Positivity
Back to coefficients of Ehrhart polynomials Federico Castillo UC Davis Ehrhart Positivity
Back to coefficients of Ehrhart polynomials What is known? Federico Castillo UC Davis Ehrhart Positivity
Back to coefficients of Ehrhart polynomials What is known? 1 The leading coefficient of i ( P , m ) is the volume vol ( P ) of P . Federico Castillo UC Davis Ehrhart Positivity
Back to coefficients of Ehrhart polynomials What is known? 1 The leading coefficient of i ( P , m ) is the volume vol ( P ) of P . 2 The second coefficient equals 1 / 2 of the sum of the normalized volumes of each facet. Federico Castillo UC Davis Ehrhart Positivity
Back to coefficients of Ehrhart polynomials What is known? 1 The leading coefficient of i ( P , m ) is the volume vol ( P ) of P . 2 The second coefficient equals 1 / 2 of the sum of the normalized volumes of each facet. 3 The constant term of i ( P , m ) is always 1 . Federico Castillo UC Davis Ehrhart Positivity
Back to coefficients of Ehrhart polynomials What is known? 1 The leading coefficient of i ( P , m ) is the volume vol ( P ) of P . 2 The second coefficient equals 1 / 2 of the sum of the normalized volumes of each facet. 3 The constant term of i ( P , m ) is always 1 . No simple forms known for other coefficients for general polytopes. Federico Castillo UC Davis Ehrhart Positivity
Back to coefficients of Ehrhart polynomials What is known? 1 The leading coefficient of i ( P , m ) is the volume vol ( P ) of P . 2 The second coefficient equals 1 / 2 of the sum of the normalized volumes of each facet. 3 The constant term of i ( P , m ) is always 1 . No simple forms known for other coefficients for general polytopes. Warning It is NOT even true that all the coefficients are positive. For example, for the polytope P with vertices ( 0 , 0 , 0 ) , ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) and ( 1 , 1 , 13 ) , its Ehrhart polynomial is i ( P , n ) = 13 6 n 3 + n 2 − 1 6 n + 1 . Federico Castillo UC Davis Ehrhart Positivity
General philosophy. They are related to volumes. Federico Castillo UC Davis Ehrhart Positivity
Ehrhart Positivity Main Definition. We say an integral polytope is Ehrhart positive (or just positive for this talk) if it has positive coefficients in its Ehrhart polynomial. Federico Castillo UC Davis Ehrhart Positivity
Ehrhart Positivity Main Definition. We say an integral polytope is Ehrhart positive (or just positive for this talk) if it has positive coefficients in its Ehrhart polynomial. In the literature, different techniques have been used to proved positivity. Federico Castillo UC Davis Ehrhart Positivity
Example I Polytope: Standard simplex. Federico Castillo UC Davis Ehrhart Positivity
Example I Polytope: Standard simplex. Reason: Explicit verification. Federico Castillo UC Davis Ehrhart Positivity
Standard simplex. In the case of ∆ d = { x ∈ R d + 1 : x 1 + x 2 + · · · + x d + 1 = 1 , x i ≥ 0 } , Federico Castillo UC Davis Ehrhart Positivity
Standard simplex. In the case of ∆ d = { x ∈ R d + 1 : x 1 + x 2 + · · · + x d + 1 = 1 , x i ≥ 0 } , It can be computed that its Ehrhart polynomial is � m + d � . d (Notice how simple this h ∗ vector is). Federico Castillo UC Davis Ehrhart Positivity
Standard simplex. In the case of ∆ d = { x ∈ R d + 1 : x 1 + x 2 + · · · + x d + 1 = 1 , x i ≥ 0 } , It can be computed that its Ehrhart polynomial is � m + d � . d (Notice how simple this h ∗ vector is). More explicitly we have � m + d � = ( m + d )( m + d − 1 ) · · · ( m + 1 ) d d ! Federico Castillo UC Davis Ehrhart Positivity
Standard simplex. In the case of ∆ d = { x ∈ R d + 1 : x 1 + x 2 + · · · + x d + 1 = 1 , x i ≥ 0 } , It can be computed that its Ehrhart polynomial is � m + d � . d (Notice how simple this h ∗ vector is). More explicitly we have � m + d � = ( m + d )( m + d − 1 ) · · · ( m + 1 ) d d ! which expands positively in powers of m . Federico Castillo UC Davis Ehrhart Positivity
Hypersimplices. In the case of ∆ d + 1 , k = conv { x ∈ { 0 , 1 } d + 1 : x 1 + x 2 + · · · + x d + 1 = k } , Federico Castillo UC Davis Ehrhart Positivity
Hypersimplices. In the case of ∆ d + 1 , k = conv { x ∈ { 0 , 1 } d + 1 : x 1 + x 2 + · · · + x d + 1 = k } , it can be computed that its Ehrhart polynomial is Federico Castillo UC Davis Ehrhart Positivity
Hypersimplices. In the case of ∆ d + 1 , k = conv { x ∈ { 0 , 1 } d + 1 : x 1 + x 2 + · · · + x d + 1 = k } , it can be computed that its Ehrhart polynomial is d + 1 � d + 1 �� d + 1 + mk − ( m + 1 ) i − 1 � � ( − 1 ) i i d i = 0 Federico Castillo UC Davis Ehrhart Positivity
Hypersimplices. In the case of ∆ d + 1 , k = conv { x ∈ { 0 , 1 } d + 1 : x 1 + x 2 + · · · + x d + 1 = k } , it can be computed that its Ehrhart polynomial is d + 1 � d + 1 �� d + 1 + mk − ( m + 1 ) i − 1 � � ( − 1 ) i i d i = 0 Not clear if the coefficients are positive. Federico Castillo UC Davis Ehrhart Positivity
Example II Polytope: Crosspolytope Federico Castillo UC Davis Ehrhart Positivity
Example II Polytope: Crosspolytope Reason: Roots have negative real part. Federico Castillo UC Davis Ehrhart Positivity
Crosspolytope. In the case of the crosspolytope: ♦ d = conv {± e i : 1 ≤ i ≤ d } , It can be computed that its Ehrhart polynomial is d � d �� m � � 2 k , k k k = 0 which is not clear if it expands positively in powers of m . Federico Castillo UC Davis Ehrhart Positivity
Crosspolytope. However Federico Castillo UC Davis Ehrhart Positivity
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