aa r X i v : . [ m a t h . C O ] O c t SIMPLICES FOR NUMERAL SYSTEMS
LIAM SOLUS
Abstract.
The family of lattice simplices in R n formed by the convex hull ofthe standard basis vectors together with a weakly decreasing vector of negativeintegers include simplices that play a central role in problems in enumerativealgebraic geometry and mirror symmetry. From this perspective, it is use-ful to have formulae for their discrete volumes via Ehrhart h ∗ -polynomials.Here we show, via an association with numeral systems, that such simplicesyield h ∗ -polynomials with properties that are also desirable from a combinato-rial perspective. First, we identify n -simplices in this family that associate viatheir normalized volume to the n th place value of a positional numeral system.We then observe that their h ∗ -polynomials admit combinatorial formula viadescent-like statistics on the numeral strings encoding the nonnegative integerswithin the system. With these methods, we recover ubiquitous h ∗ -polynomialsincluding the Eulerian polynomials and the binomial coefficients arising fromthe factoradic and binary numeral systems, respectively. We generalize the bi-nary case to base- r numeral systems for all r ≥
2, and prove that the associated h ∗ -polynomials are real-rooted and unimodal for r ≥ n ≥ Introduction
We consider the family of simplices defined by the convex hull∆ (1 ,q ) := conv( e , . . . , e n , − q ) ⊂ R n , where e , . . . , e n denote the standard basis vectors in R n , and q = ( q , q , . . . , q n ) isany weakly increasing sequence of n positive integers. A convex polytope P ⊂ R n with vertices in Z n (i.e. a lattice polytope ) is called reflexive if its polar body P ◦ := { y ∈ R n : y T x ≤ x ∈ P } ⊂ R n is also a lattice polytope. In the case of ∆ (1 ,q ) , the reflexivity condition is equivalentto the arithmetic condition q i | X j = i q j for all j ∈ [ n ] , which was used to classify all reflexive n -simplices in [7]. Reflexive simplices of theform ∆ (1 ,q ) have been studied extensively from an algebro-geometric perspectivesince they exhibit connections with string theory that yield important results inenumerative geometry [8]. These connections result in an explicit formula for allGromov-Witten invariants which, in turn, encode the number of rational curvesof degree d on the quintic threefold [6]. This observation lead to the developmentof the mathematical theory of mirror symmetry in which reflexive polytopes andtheir polar bodies play a central role. In particular, the results of [6] are intimatelytied to the fact that ∆ (1 ,q ) for q = (1 , , , ∈ R contains significantly less latticepoints (i.e. points in Z n ) than its polar body. Date : October 5, 2017.
LIAM SOLUS
As a result of this connection, the lattice combinatorics and volume of the sim-plices ∆ (1 ,q ) have been studied from the perspective of geometric combinatorics interms of their (Ehrhart) h ∗ -polynomials. The Ehrhart function of a d -dimensionallattice polytope P is the function i ( P ; t ) := | tP ∩ Z n | , where tP := { tp : p ∈ P } denotes the t th dilate of the polytope P for t ∈ Z ≥ . It is well-known [9] that i ( P ; t )is a polynomial in t of degree d , and the Ehrhart series of P is the rational function X t ≥ i ( P ; t ) z t = h ∗ + h ∗ z + · · · + h ∗ d z d (1 − z ) d +1 , where the coefficients h ∗ , h ∗ , . . . , h ∗ d are all nonnegative integers [19]. The poly-nomial h ∗ ( P ; z ) := h ∗ + h ∗ z + · · · + h ∗ d z d is called the h ∗ -polynomial of P . The h ∗ -polynomial of P encodes the typical Euclidean volume of P in the sense that d ! vol( P ) = h ∗ ( P ; 1). It also encodes the number of lattice points in P since h ∗ = | P ∩ Z n | − d − h ∗ , . . . , h ∗ d ∈ Z ≥ ,distributional properties of these coefficients is a popular research topic. Let p := a + a z + · · · + a d z d be a polynomial with nonnegative integer coefficients. Thepolynomial p is called symmetric if a i = a d − i for all i ∈ [ d ], it is called unimodal ifthere exists an index j such that a i ≤ a i +1 for all i < j and a i ≥ a i +1 for all i ≥ j ,it is called log-concave if a i ≥ a i − a i +1 for all i ∈ [ d ], and it is called real-rooted if all of its roots are real numbers. An important result in Ehrhart theory statesthat P ⊂ R n is reflexive if and only if h ∗ ( P ; x ) is symmetric of degree n [12]. It iswell-known that if p is real-rooted then it is log-concave, and if all a i > h ∗ -polynomials.The distributional (and related algebraic) properties of the h ∗ -polynomials forthe simplices ∆ (1 ,q ) were recently studied in terms of the arithmetic structureof q [3, 4]. In particular, [4, Theorem 2.5] provides an arithmetic formula for h ∗ (∆ (1 ,q ) ; z ) in terms of q which the authors use to prove unimodality of the h ∗ -polynomial in some special cases. On the other hand, the literature lacks examplesof ∆ (1 ,q ) admitting combinatorial formula for their h ∗ -polynomials. Thus, whilethe simplices ∆ (1 ,q ) constitute a class of convex polytopes fundamental in algebraicgeometry, we would like to observe that they are of interest from a combinatorialperspective as well. To demonstrate that this is in fact the case, we will identifysimplices ∆ (1 ,q ) whose h ∗ -polynomials are well-known in combinatorics and admitthe desirable distributional properties mentioned above. We observe that simpliceswithin this family can be associated in a natural way to numeral systems andthe h ∗ -polynomials of such simplices admit a combinatorial interpretation in termsof descent-type statistics on the numeral representations of nonnegative integerswithin these systems. Most notably, we find such ubiquitous generating polyno-mials as the Eulerian polynomials and the binomial coefficients arising from the factoradic and binary numeral systems, respectively. For these examples, we alsofind that the combinatorial interpretation of h ∗ (∆ (1 ,q ) ; z ) is closely tied to that of q . We then generalize the simplices arising from the binary system to simplicesassociated to any base- r numeral system for r ≥
2, all of whose h ∗ -polynomialsadmit a combinatorial interpretation in terms of the base- r representations of thenonnegative integers. Finally, we show that, even though these simplices are not allreflexive, their h ∗ -polynomials are always real-rooted, log-concave, and unimodal. IMPLICES FOR NUMERAL SYSTEMS 3
The remainder of the paper is outlined as follows. In Section 2, we review thebasics of positional numeral systems and outline our notation. In Section 3, wedescribe when a numeral system associates to a family of reflexive simplices ofthe form ∆ (1 ,q ) , one in each dimension n ≥
1. We then show that the n -simplexfor the factoradic numeral system has the ( n + 1) st Eulerian polynomial as its h ∗ -polynomial. We also prove the n -simplex for the binary numeral system has h ∗ -polynomial (1 + x ) n . In Section 4, we generalize the binary case to base- r numeralsystems for r ≥
2. We provide a combinatorial formula for these h ∗ -polynomialsand prove they are real-rooted, log-concave, and unimodal for r ≥ n ≥ Numeral Systems
The typical (positional) numeral system is a method for expressing numbersthat can be described as follows. A numeral is a sequence of nonnegative integers η := η n − η n − . . . η η , and the location of η i in the string is called place i . The digits are the numbers allowable in each place, and the i th radix (or base) is thenumber of digits allowed in place i . A numeral system is a sequence of positiveintegers a = ( a n ) ∞ n =0 satisfying a := 1 < a < a < · · · , which we call place values .To see why a yields a system for expressing nonnegative integers, let b ∈ Z ≥ andlet n be the smallest integer so that a n > b . Dividing b by a n − and iterating gives b = q n − a n − + r n − ≤ r n − < a n − r n − = q n − a n − + r n − ≤ r n − < a n − ... ... r i +1 = q i a i + r i ≤ r i < a i ... ... r = q a + r ≤ r < a r = q a . Denoting b a ( i ) := q i for all i , it follows that b = b a ( n − a n − + b a ( n − a n − + · · · + b a (1) a + b a (0) a . (1)On the other hand, if b = P n − i =0 β i a i where P ij =0 β i a i < a i +1 for every i ≥ b a ( i ) = β i for every i . In particular, the representation of b in equation(1) is unique (see for instance [11, Theorem 1]). Thus, the representation of thenonnegative integer b in the numeral system a is the numeral b a := b a ( n − b a ( n − · · · b a (1) b a (0) . An important family of numeral systems we will consider are the mixed radixsystems. A numeral system a = ( a n ) ∞ n =0 is mixed radix if there exists a sequenceof integers ( c n ) ∞ n =0 with c := 1 and c n > n > c c · · · c n = a n for all n ≥
0. In this case, c n +1 is the n th radix of the system a . Example 2.1 (Numeral systems) . The following are examples of numeral systems.(1) The binary numbers are the numeral system a = (2 n ) ∞ n =0 . The radix ofplace i is 2 for every i since each place assumes only digits 0 or 1. Thebinary system is mixed radix with sequence of radices c = (1 , , , , . . . ).The numeral representation of the number 102 in this system is 1100110. LIAM SOLUS (2) The ternary (base-3) numeral system is the system a = (3 n ) ∞ n =0 . The radixof place i is 3 for every i since each place assumes only digits 0 , c = (1 , , , , , . . . ). The numeral representation of the number 102 in theternary system is 10210.(3) An example of a numeral system that is not mixed radix is the system a = ( F n +1 ) ∞ n =0 , where a n = F n +1 is the ( n + 1) st Fibonacci number. Thissequence is not mixed radix since there exist prime Fibonacci numbers otherthan 2. The radix of every place is 2 since 2 F n +1 = F n +1 + F n > F n +1 forall n , and so each place assumes digits 0 or 1. The numeral representationof the number 102 in this system is 1000100000.3. Reflexive Numeral Systems
In this section, we demonstrate a method for attaching an n -simplex of the form∆ (1 ,q ) to the n th place value a n of a numeral system a such that a n = n ! vol(∆ (1 ,q ) ),i.e. the normalized volume of P . For certain families of numeral systems, which wecall reflexive numeral systems , these simplices can be chosen so that they are all re-flexive. We show that the factoradic and binary numeral systems are reflexive, andrecover the h ∗ -polynomials of their associated simplices. We also discuss the rela-tionship between reflexive numeral systems and the mixed radix numeral systems,and the geometric relationship between the factoradic n -simplex and the s -lecturehall n -simplex with the same h ∗ -polynomial, i.e. the n th Eulerian polynomial.
Definition 3.1.
A numeral system a = ( a n ) ∞ n =0 is called a reflexive (numeral) sys-tem if there exists an increasing sequence of positive integers d = ( d n ) ∞ n =0 satisfyingthe following properties:(1) d i | a n for all 0 ≤ i ≤ n − n ≥
1, and(2) 1 + P n − i =0 a n d i = a n for all n ≥ d is called a divisor system (for a) . Example 3.1 (A reflexive numeral system) . Recall from Example 2.1 that thebinary numeral system is given by a = ( a n ) ∞ n =0 := (2 n ) ∞ n =0 . The binary system a admits the divisor system d = ( d n ) ∞ n =0 := (2 n +1 ) ∞ n =0 . This is because for all n ≥ i +1 | n for all 0 ≤ i ≤ n −
1, and1 + n − X i =0 a n d i = 1 + n − X i =0 n i +1 = 1 + n − X i =0 i = 2 n = a n . Thus, the binary numeral system a is a reflexive system with divisor system d .Furthermore, in Theorem 3.6 we will prove that h ∗ (∆ (1 ,q ) ; z ) = (1 + z ) n . Recall that we would like to associate an n -simplex ∆ (1 ,q ) to the n th place value a n of a numeral system a by the relation a n = n ! vol(∆ (1 ,q ) ). In [4, 17] is it shownthat the normalized volume of ∆ (1 ,q ) (i.e. the value n ! vol(∆ (1 ,q ) )) is 1+ q + · · · + q n .When a is a reflexive system with divisor system d , condition (1) tells us that a n d i is a positive integer for all n ≥ ≤ i ≤ n −
1, and condition (2) tells us thatthe n -simplex ∆ (1 ,q ) with q := (cid:18) a n d n − , a n d n − , . . . , a n d (cid:19) ∈ Z n> will have the desired normalized volume. Moreover, it turns out the h ∗ -polynomialof this n -simplex can be computed in a recursive fashion in terms of the numeralrepresentations of the first a n nonnegative integers. This recursive formula is pre-sented in Proposition 3.2, but it is essentially due to the following theorem of [4]. Theorem 3.1. [4, Theorem 2.5]
The h ∗ -polynomial for the n -simplex ∆ (1 ,q ) for q = ( q , . . . , q n ) is h ∗ (∆ (1 ,q ) ; z ) = q + q + ··· + q n X b =0 z ω ( b ) , where ω ( b ) = b − n X i =1 (cid:22) q i b q + q + · · · + q n (cid:23) . Proposition 3.2.
Let a = ( a n ) ∞ n =0 be a reflexive system that admits a divisorsystem d = ( d n ) ∞ n =0 . Then for q := (cid:18) a n d n − , a n d n − , . . . , a n d (cid:19) the reflexive n -simplex ∆ (1 ,q ) ⊂ R n has h ∗ -polynomial h ∗ (∆ (1 ,q ) ; z ) = P a n − b =0 z ω ( b ) , where ω ( b ) = ω ( b ′ ) + b a ( n − − (cid:22) bd n − (cid:23) , and b ′ := b − b a ( n − a n − .Proof. Applying Theorem 3.1 to our particular case, we can simplify the formulafor ω ( b ) as follows: ω ( b ) = b − n − X i =0 (cid:22) bd i (cid:23) , = b a ( n − a n − + b ′ − n − X i =0 (cid:22) b a ( n − a n − + b ′ d i (cid:23) − (cid:22) bd n − (cid:23) , = b a ( n − a n − + b ′ − b a ( n − n − X i =0 a n − d i ! − n − X i =0 (cid:22) b ′ d i (cid:23) − (cid:22) bd n − (cid:23) , = ω ( b ′ ) + b a ( n − − (cid:22) bd n − (cid:23) . (cid:3) We note that not every mixed radix system is a reflexive system. However, if amixed radix system is reflexive, then its corresponding divisor system is unique. If a is mixed radix then a n = c · · · c n where c = ( c n ) ∞ n =0 is its sequence of radices.Using this fact and part (2) of Definition 3.1, we deduce the following proposition. Proposition 3.3. If a = ( a n ) ∞ n =0 is a reflexive mixed radix system with sequenceof radices c = ( c n ) ∞ n =0 then its corresponding divisor system d is unique and d = ( d n ) ∞ n =0 = (cid:18) a n +1 c n +1 − (cid:19) ∞ n =0 . LIAM SOLUS
Example 3.2 (Reflexive and non-reflexive mixed radix systems) . Recall from Ex-ample 3.1 that the binary numeral system a = (2 n ) ∞ n =0 is a reflexive system withdivisor system d = (2 n +1 ) ∞ n =0 . Since a is also a mixed radix system with sequenceof radices c = (1 , , , , , . . . ), then we can apply Proposition 3.3 to recover thedivisor system d as the unique divisor system for a .It is also important to notice that there are mixed radix systems that are notreflexive. For example, consider the numeral system a = (2 n · n !) ∞ n =0 whose n th placevalue is the order of the hyperoctahedral group. Then a is reflexive with sequenceof radices c = (1 , , , . . . , n, . . . ). However, it follows from Proposition 3.3 that a has no divisor system since there are infinitely many times when n +1 ( n +1)!2( n +1) − is notan integer. For instance, every prime is expressible as 2( n + 1) − n , andthis prime will never be a factor of 2 n +1 ( n + 1)!.Although identifying a divisor system for a given numeral system is generallya nontrivial problem, Proposition 3.3 makes it easier in the case of mixed radixsystems, and it provides a quick check to deduce if the system is reflexive. Moreover,when a mixed radix system is reflexive, the resulting simplices ∆ (1 ,q ) appear to havewell-known h ∗ -polynomials with a combinatorial interpretation closely related to acombinatorial interpretation of q . We now give two examples of this phenomenon.3.1. The factoradics and the Eulerian polynomials.
In this subsection, westudy the numeral system a = ( a n ) ∞ n =0 := (( n + 1)!) ∞ n =0 , which is commonly calledthe factoradics . By Proposition 3.3, we see that the factoradics are reflexive withdivisor system d = ( d n ) ∞ n =0 := (( n +1)!+ n !) ∞ n =0 . We will see that the q -vectors givenby d admit a combinatorial interpretation in terms of descents, and the resultingsimplices ∆ (1 ,q ) have h ∗ -polynomials the Eulerian polynomials. For π ∈ S n , we letDes( π ) := { i ∈ [ n −
1] : π i +1 > π i } , des( π ) := | Des( π ) | , maxDes( π ) := ( max { i ∈ [ n −
1] : i ∈ Des( π ) } for π = 12 · · · n ,0 for π = 12 · · · n .We then consider the pair of polynomial generating functions A n ( z ) := X π ∈ S n z des( π ) and B n ( z ) := X π ∈ S n z maxDes( π ) , where A n ( z ) is the well-known n th Eulerian polynomial . The polynomial B n ( z ) isa different generating function for permutations that satisfies a recursion describedin Lemma 3.4. In the following, we let A ( n, k ) and B ( n, k ) denote the coefficientof z k in A n ( z ) and B n ( z ), respectively. Lemma 3.4.
For each n ∈ Z > we have that B ( n,
0) = 1 , B ( n,
1) = n − , and for k > B ( n, k ) = ( n ) k − ( n − k ) = nB ( n − , k − . Moreover, for the polynomials B n ( z ) , we have the recursive expression B n ( z ) = 1 − z + nzB n − ( z ) , ( n > IMPLICES FOR NUMERAL SYSTEMS 7 and the closed-form expression B n ( z ) = 1 + n − X k =1 n !( n − k )! + ( n − k − z k . Proof.
By the definition of B n ( z ) we can see that B ( n,
0) = 1 and B ( n,
1) = n − n ∈ Z > . To see that B ( n, k ) = ( n ) k − ( n − k ) for k >
1, notice first thatthe falling factorial ( n ) k − counts the number of ways to pick the first k − π = π π · · · π n . Multiplying this value by ( n − k ) accounts for the fact thatthere are ( n − k ) remaining choices for the k th letter such that π k > π k +1 . Theremaining ( n − k ) letters of the permutation are then arranged in increasing order,and so maxDes( π ) = k . The recursion for B ( n, k ) for k > B n ( z ) for n > B n ( z )is immediate from the closed forms presented for the coefficients B ( n, k ) for k ≥ n ! = ( n − n − n − (cid:3) We now show that the sequence of coefficients for the nonconstant terms of B n ( z )is a vector q for which the n -simplex ∆ (1 ,q ) has h ∗ -polynomial A n ( z ); thereby offer-ing, in a sense, a geometric transformation between the two generating polynomials.Recall that for two integer strings η := η η · · · η n and µ := µ µ · · · µ n the string η is lexicographically larger than the string µ if and only if the leftmost nonzeronumber in the string ( η − µ )( η − µ ) · · · ( η n − µ n ) is positive. For 0 ≤ b < n !, thefactoradic representation of b , denoted b ! := b ! ( n − b ! ( n − · · · b ! (1) b ! (0) , is knownas the Lehmer code of the b th largest permutation of [ n ] under the lexicographicordering [15]. In particular, if we let π ( b ) denote the b th largest permutation of [ n ]under the lexicographic ordering, then for all 0 ≤ i < nb ! ( i ) = (cid:12)(cid:12)(cid:12) { ≤ j < i : π ( b ) n − i > π ( b ) n − j } (cid:12)(cid:12)(cid:12) . It is straightforward to check that b ! ( i ) > b ! ( i + 1) if and only if n − i ∈ Des( π ( b ) ).Thus, counting descents in π ( b ) is equivalent to counting descents in the factoradicrepresentation of the integer b . This fact allows for the following theorem. Theorem 3.5.
The factoradic numeral system a = ( a n ) ∞ n =0 = (( n + 1)!) ∞ n =0 admitsthe divisor system d = ( d n ) ∞ n =0 = (( n + 1)! + n !) ∞ n =0 for which the reflexive simplex ∆ (1 ,q ) ⊂ R n with q := ( B ( n + 1 , , B ( n + 1 , , . . . , B ( n + 1 , n )) has h ∗ -polynomial h ∗ (∆ (1 ,q ) ; z ) = A n +1 ( z ) . Proof.
First notice that d = (( n + 1)! + n !) ∞ n =0 is in fact a divisor system for a = (( n + 1)!) ∞ n =0 by Lemma 3.4. We now show, via induction on n , that for allintegers b = 0 , , . . . , n ! − ω ( b ) = des( π ( b ) ) , where π ( b ) denotes the b th largestpermutation of [ n ]. Since the base case ( n = 1) is clear, we assume the result holdsfor n −
1. For convenience, we reindex a = ( n !) ∞ n =1 and d = ( n ! + ( n − ∞ n =1 . LIAM SOLUS
Then by Proposition 3.2 we know that for b = 0 , , . . . , n ! ω ( b ) = ω ( b ′ ) + b ! ( n − − (cid:22) bd n − (cid:23) , = ω ( b ′ ) + b ! ( n − − (cid:22) ( n − b ! ( n − n − b ′ n ! (cid:23) , = ω ( b ′ ) + b ! ( n − − (cid:22) b ! ( n − − (cid:18) b ! ( n − n − − ( n − b ′ n ! (cid:19)(cid:23) , = ω ( b ′ ) + (cid:24) b ! ( n − n − − ( n − b ′ n ! (cid:25) . (2)With equation (2) in hand, we now consider a few cases. First, notice that if b ! ( n −
1) = 0 then ω ( b ) = ω ( b ′ ), and the result follows from the inductive hypothesis.This is because 0 ≤ b n − ≤ ( n − −
1. So whenever b ! ( n −
1) = 0 then (cid:24) b ! ( n − n − − ( n − b ′ n ! (cid:25) = (cid:24) − ( n − b ′ n ! (cid:25) = 0 . Suppose now that 0 < b ! ( n − < n . Then π ( b ) satisfies π ( b )1 = b ! ( n −
1) + 1, andwe consider the following three cases.First, if b ′ = 0, then since 0 < b ! ( n − < n , we have that ω ( b ) = ω ( b ′ ) + 1, andthe result follows from the inductive hypothesis. Second, suppose that b ′ = 0 andthat 0 < b ! ( n − n − − ( n − b ′ . Then since 0 < ( n − b ′ ≤ b ! ( n − n − < n !,we know that (cid:24) b ! ( n − n − − ( n − b ′ n ! (cid:25) = 1 . Thus, ω ( b ) = ω ( b ′ ) + 1. Since the first b ! ( n − n − n ] with π = b ! ( n −
1) + 1 satisfy π > π , the result follows from the inductive hypothesis.Finally, if b ′ = 0 and 0 ≤ ( n − b ′ − b ( n − n − < ( n − b ′ ≤ ( n − n − − < n !, we have that (cid:24) b ! ( n − n − − ( n − b ′ n ! (cid:25) = − (cid:22) ( n − b ′ − b ! ( n − n − n ! (cid:23) = 0 . Thus, ω ( b ) = ω ( b ′ ). Since the last ( n − − b ! ( n − n − n ]satisfying π = b ! ( n −
1) + 1 satisfy π < π , the result follows from the inductivehypothesis, completing the proof. (cid:3) Remark . Let F n denote the n -simplexdescribed in Theorem 3.5. To the best of the author’s knowledge, the only reflexive n -simplex ∆ with h ∗ (∆; z ) = A n +1 ( z ), other than F n , is the s -lecture hall simplex P (2 , ,...,n +1) n := (cid:26) x ∈ R n : 0 ≤ x ≤ x ≤ · · · ≤ x n n + 1 ≤ (cid:27) . Using the classification of [7], we can deduce that F n and P (2 , ,...,n +1) n define distincttoric varieties in the following sense: For a lattice n -simplex ∆ ⊂ R n containing theorigin in its interior, [7, Definition 2.3] assigns a weight q := ( q , . . . , q n ) ∈ Z n +1 > anda factor λ := gcd( q , . . . , q n ). We say that ∆ is of type ( q red , λ ), where q red := λ q .When λ = 1, the toric variety of ∆ is the weighted projective space P ( q red ), andwhen λ >
1, it is a quotient of P ( q red ) by the action of a finite group of index λ . IMPLICES FOR NUMERAL SYSTEMS 9
It follows from our construction that F n has factor 1, and so its toric variety is theweighted projective space P ( B ( n )), where B ( n ) := (1 , B ( n + 1 , , . . . , B ( n + 1 , n )).On the other hand, for n > P (2 , ,...,n +1) n empirically exhibits factor λ = n !lcm(1 , , . . . , n ) , (see sequence [18, A025527]), and q red = B ( n ). Thus, F n and P (2 , ,...,n +1) n definedistinct toric varieties in terms of the classification of [7]. Moreover, F n appears tobe the only known weighted projective space with Eulerian h ∗ -polynomial.A second way to see that F n and P (2 , ,...,n +1) n define distinct toric varieties is tonote that F n is not self-dual for n ≥
3; that is, for n ≥
3, the polar body of F n is not F n itself up to a translation and/or a rotation. On the other hand, [13] proves that P (2 , ,...,n +1) n is a self-dual polytope. Thus, from a discrete geometric perspective,this allows us to see that F n and P (2 , ,...,n +1) n define distinct toric varieties. Remark . We also note that the
Type B Eulerianpolynomials cannot arise from a numeral system since the sequence a := (2 n · n !) ∞ n =0 is a mixed radix system that is not reflexive (see Example 3.2).3.2. The binary numbers and binomial coefficients.
Using Proposition 3.3,we can see that the binary numeral system a = ( a n ) ∞ n =0 = (2 n ) ∞ n =0 is also reflexive(see Example 3.2). Here, the h ∗ -polynomial of the resulting n -simplices are given bycounting the number of 1’s in the binary representation of the first 2 n nonnegativeintegers. In the following, we let supp ( b ) denote the number of nonzero digits inthe binary representation b := b ( n − b ( n − · · · b (0) of the integer b . Theorem 3.6.
The binary numeral system a = ( a n ) ∞ n =0 = (2 n ) ∞ n =0 admits thedivisor system d = ( d n ) ∞ n =0 = (2 n +1 ) ∞ n =0 for which the reflexive simplex ∆ (1 ,q ) ⊂ R n with q := (cid:0) , , , , . . . , n − (cid:1) has h ∗ -polynomial h ∗ (∆ (1 ,q ) ; z ) = n − X b =0 z supp ( b ) = (1 + z ) n . Proof.
To prove the result we show that ω ( b ) = supp ( b ) for all b = 0 , , , . . . , n − n . For the base case, we take n = 1. By [4, Theorem 2.5] we havethat h ∗ (∆ (1 ,q ) ; z ) = z ω (0) + z ω (1) where ω (0) = 0 = supp (0) , and ω (1) = 1 = supp (1) . For the inductive step,suppose that ω ( b ) = supp ( b ) for all b = 0 , , , . . . , n − . Then by Proposition 3.2we have that ω ( b ) = ω ( b ′ ) + b ( n − − (cid:22) bd n − (cid:23) = ω ( b ′ ) + b ( n − , where the last equality holds since 0 ≤ b < n . Sincesupp ( b ) = supp ( b ′ ) + b ( n − , the result follows by the inductive hypothesis. Finally, the fact that h ∗ (∆ (1 ,q ) ) =(1 + z ) n follows from [11, Theorem 1] and the fact that any (0 , n is a valid binary representation of a nonnegative integer less than 2 n . (cid:3) Similar to the factoradics, the q -vector in Theorem 3.6 can be viewed as thecoefficients of a “max-descent” polynomial for binary strings of length n . Namely, q i is the number of binary strings η n − η n − · · · η with right-most nonzero digit η i .Analogously, Theorem 3.6 provides a geometric transformation between these twogenerating polynomials for counting binary strings in terms of their nonzero entries.Proposition 3.3 implies that is the only reflexive base- r numeral system for r ≥ r generalization of The-orem 3.6 that results in simplices with symmetric h ∗ -polynomials. On the otherhand, there is a generalization that preserves other desirable properties of the h ∗ -polynomial (1 + x ) n , including real-rootedness and unimodality. This is the focusof the next section.4. The Positional Base- r Numeral Systems
For r ≥
2, consider the base- r numeral system, a := ( r n ) ∞ n =0 . Just as in thebase-2 case, we denote the base- r representation of an integer b ∈ Z ≥ by b r := b r ( n − b r ( n − · · · b r (0) . We will now study a generalization of the n -simplices from Theorem 3.6 for r ≥ h ∗ -polynomials preserve many of the nice properties of the r = 2 case, includ-ing real-rootedness and unimodality. Our generalization is motivated as follows:In order to simplify the formula for ω ( b ) to the desired state in the proofs ofTheorem 3.5 and Theorem 3.6 respectively, we required the identities1 + n − X k =0 k · k ! = n ! and 1 + n − X k =0 k = 2 n . Notice that these identities are different than the one requested in Definition 3.1(2) to certify reflexivity of a numeral system. In fact, it follows from [11, Theorem2] that any mixed radix system a = ( a n ) ∞ n =0 with sequence of radices c = ( c n ) ∞ n =1 satisfies the identity 1 + n − X k =0 ( c k +1 − a k = a n . In the case of base- r numeral systems, this identity yields a natural generalizationof Theorem 3.6. For two integers r ≥ n ≥
1, we define the base- r n -simplex to be the n -simplex B ( r,n ) := ∆ (1 ,q ) ⊂ R n for q := (cid:0) ( r − , ( r − r, ( r − r , . . . , ( r − r n − (cid:1) . In the following, we show that, while symmetry of h ∗ ( B ( r,n ) ; z ) does not hold for r >
2, many of the nice properties of h ∗ ( B (2 ,n ) ; z ) carry over to this more gen-eral family. In Subsection 4.1 we prove that h ∗ ( B ( r,n ) ; z ) admits a combinatorialinterpretation in terms of a descent-like statistic applied to the nonzero digits ofthe base- r representations of the nonnegative integers. Then, in Subsection 4.2, weprove that h ∗ ( B ( r,n ) ; z ) is real-rooted and unimodal for all r ≥ n ≥ A descent-like statistic.
We now define a descent-like statistic on the base- r representations of nonnegative integers that we then use to give a combinatorialinterpretation of the h ∗ -polynomial of B ( r,n ) for r ≥ n ≥
1. Given two indices i ≥ j and an integer b ∈ Z ≥ , we can think of the integer quantity b r ( i ) − b r ( j ) asthe height of the index i “above” the index j in the string b r . Of course, a negative IMPLICES FOR NUMERAL SYSTEMS 11 b base-4 numeral Supp ( b ) awheight of numerals in Supp ( b ) nasc ( b )19 103 { , } awheight(0) = 1 2awheight(2) = 122 112 { , , } awheight(0) = 1 1awheight(1) = − − { , , } awheight(0) = 1 2awheight(1) = 0awheight(2) = − Table 1.
The statistic nasc r ( b ) for some integers b with respectto the base-4 numeral system a = (4 n ) ∞ n =0 and n = 3.height simply means we think of i as “below” j in b r . We define the (averageweighted) height of an index i > b r to beawheight( i ) := 1 i i − X j =0 ( b r ( i ) − b r ( j )) r j , and awheight(0) := ( b r (0) = 0 , b r (0) = 0 . In a sense, this statistic measures the height of i above the remaining substring of b r where the value of the height of an index closer to i (in absolute value) is higher.When awheight( i ) is nonnegative, we can think of i as being at least as high asthe remaining portion of the string, and so we say that i is a nonascent of b r if0 ≤ awheight( i ) . Define the support of b to be the setSupp r ( b ) := { i ∈ Z ≥ : b r ( i ) = 0 } and let supp r ( b ) := | Supp r ( b ) | . We then consider the collection of indicesNasc r ( b ) := { i ∈ Supp r ( b ) : 0 ≤ awheight( i ) } , and we let nasc r ( b ) := | Nasc r ( b ) | . Example 4.1 (Computing nasc r ( b ) for base-4 numerals) . The base-4 numeral sys-tem is a = (4 n ) ∞ n =0 . The numeral representations for the first 64 nonnegativeintegers are the strings of length three b (2) b (1) b (0) in which each term b ( i ) canassume values 0 , ,
2, or 3. For example, the number b = 19 has base-4 representa-tion 103, and so the support of b is Supp ( b ) = { , } . To compute nasc ( b ) we mustcompute the (averaged weighted) height of the indices 0 and 2. Since b (0) = 3then awheight(0) = 1, andawheight(2) = 12 (cid:0) (1 − · + (1 − · (cid:1) = 1 . Thus, nasc ( b ) = 2. Table 1 presents this statistic for a couple more integers. Giventhe statistic nasc ( b ) for all integers 0 ≤ b <
64 we can compute h ∗ ( B (4 , ; z ) = X b =0 z nasc ( b ) = 1 + 19 z + 34 z + 10 x . The following theorem shows that this formula generalizes to the base- r n -simplicesfor all n ≥ r ≥ Theorem 4.1.
For two integers r ≥ and n ≥ the base- r n -simplex B ( r,n ) has h ∗ -polynomial h ∗ ( B ( r,n ) ; z ) = r n − X b =0 z nasc r ( b ) . Proof.
By [4, Theorem 2.5], it suffices to show for 0 ≤ b < r n that ω ( b ) = nasc r ( b ) . We prove this fact via induction on n . Notice first that ω ( b ) = b − n X k =1 (cid:22) ( r − br i (cid:23) . For the base, take n = 1, and notice that ω ( b ) = (cid:6) br (cid:7) , and so ω ( b ) = 0 where b = 0,and ω ( b ) = 1 for 1 ≤ b ≤ r −
1. Suppose now that the result holds for n −
1. Letting b ′ := b − b r ( n − r n − , we then observe that ω ( b ) = b − n X k =1 (cid:22) ( r − br i (cid:23) , = b − n X k =1 (cid:22) ( r − b r ( n − r n − + b ′ ) r i (cid:23) , = b − n X k =1 (cid:22) ( r − b r ( n − r n − r i (cid:23) − n X k =1 (cid:22) ( r − b ′ r i (cid:23) − (cid:22) ( r − br n (cid:23) , = ω ( b ′ ) + b r ( n − r n − − b r ( n − n − X k =0 r k ! − (cid:22) ( r − br n (cid:23) , = ω ( b ′ ) + (cid:24) rb r ( n − r n − − ( r − b ′ r n (cid:25) , Notice that rb r ( n − r n − − ( r − b ′ > b r ( n − = 0 and b ′ < b r ( n − (cid:18) r n − r − (cid:19) , n − X j =0 b r ( j ) r j < n − X j − b r ( n − r j + b r ( n − r − , − b r ( n − r − n − < n − n − X j − ( b r ( n − − b r ( j )) r j . Since 1 ≤ b r ( n − ≤ r −
1, this last inequality is equivalent to n − b r , which completes the proof. (cid:3) Real-rootedness and unimodality.
To prove real-rootedness of the h ∗ -polynomial of B ( r,n ) , we will use the well-developed theory of interlacing polynomi-als. Let f, g ∈ R [ z ] be nonzero, real-rooted polynomials, and let d := deg( f ), and c := deg( g ) denote the degree of f and g , respectively. Suppose α d ≤ · · · ≤ α ≤ α and β c ≤ · · · ≤ β ≤ β are the roots of f and g , respectively. We say that g interlaces f , written g (cid:22) f , if either d = c and β d ≤ α d ≤ · · · ≤ β ≤ α ≤ β ≤ α , IMPLICES FOR NUMERAL SYSTEMS 13 or d = c + 1 and α d +1 ≤ β d ≤ α d ≤ · · · ≤ β ≤ α ≤ β ≤ α . If all inequalities are strict, we say that g strictly interlaces f and we write g ≺ f .A sequence F m := ( f i ) mi =1 of real-rooted polynomials is called (strictly) interlac-ing if f i (strictly) interlaces f j for all 1 ≤ i < j ≤ m . Let F + m denote the spaceof all interlacing sequences F m for which f i has only nonnegative coefficients forall 1 ≤ i ≤ n . In [5], Br¨and´en characterized when a matrix G = ( G i,j ( z )) mi,i =1 ofpolynomials maps F + m to F + m . We say that such a map preserves interlacing . It pre-serves strict interlacing if it further maps strictly interlacing sequences to strictlyinterlacing sequences. In the following we use such polynomial maps to prove thereal-rootedness of h ∗ ( B ( r,n ) ; z ) for r ≥ n ≥ r ≥ n ≥ f ( r,n ) := (1 + z + z + · · · + z r − ) n . As noted in [14], for every r ≥ f ∈ R [ z ] there are uniquely determined f (0) , . . . , f ( r − ∈ R [ z ] such that f ( z ) = f (0) ( z r − ) + zf (1) ( z r − ) + · · · + z r − f ( r − ( z r − ) . So, for ℓ = 0 , , . . . , r −
2, we consider the operator h r − ,ℓ i : R [ z ] −→ R [ z ] where h r − ,ℓ i : f −→ f ( ℓ ) . The following theorem gives a second interpretation of the h ∗ -polynomial of B ( r,n ) ,now in terms of the polynomials f h r − ,ℓ i ( r,n ) . Theorem 4.2.
For two integers r ≥ and n ≥ the base- r n -simplex B ( r,n ) has h ∗ -polynomial h ∗ ( B ( r,n ) ; z ) = f h r − , i ( r,n ) + z r − X ℓ =1 f h r − ,ℓ i ( r,n ) . Proof.
To prove this result, we prove a slightly stronger statement. We will show,via induction on n , that r + r + ··· + r n − X b =0 z ω ( b ) = f h r − , i ( r,n ) , and for each i ∈ [ r − ( i +1)(1+ r + r + ··· + r n − ) X b =1+ i (1+ r + r + ··· + r n − ) z ω ( b ) = zf h r − ,r − ℓ − i ( r,n ) . For the base case, we let n = 1, and so we must verify that z ω (0) + z ω (1) = f h r − , i ( r, , and for each i ∈ [ r − z ω ( i +1) = zf h r − ,r − ℓ − i ( r, . Notice first that since n = 1, then ω ( b ) = (cid:6) br (cid:7) for all b = 0 , , . . . , r −
1, and so ω ( b ) = ( b = 0,1 if b ∈ [ r − The base case follows immediately from the fact that f ( r, = 1 + r + r + · · · + r n − . As for the inductive step, we begin by partitioning the sequence of numbers B = ( B j ) r n − j =0 := (0 , , , . . . , r n − r consecutive sequences B i = ( B i,j ) r n − − j =0 := ( B j ) ( i +1) r n − − j = ir n − for i = 0 , , . . . , r −
1. Notice that if b ∈ B i then b r ( n −
1) = i . Even more, thenumber b has the base- r representation b r ( n − b r ( n − · · · b r (1) b r (0) being the b th sequence of n digits 0 , , . . . , r − i = 1 , , . . . , r − ω ( B i,j ) = ( ω ( B ,j ) + 1 if j ≤ r + r + · · · + r n − , ω ( B ,j ) if j > r + r + · · · + r n − . (3)This is because the base- r representation of b = i (1+ r + r + · · · + r n − ) is b r = ii · · · i .Combining the observation in equation (3) with the inductive hypothesis, we thenhave that r + r + ··· + r n − X b =0 z ω ( b ) = r n − − X b =0 z ω ( b ) + r + r + ··· + r n − X b = r n − z ω ( b ) , = h ∗ ( B ( r,n − ; z ) + zf h r − , i ( r,n − , = f h r − , i ( r,n − + z r − X ℓ =1 f h r − ,ℓ i ( r,n − + zf h r − , i ( r,n − , = f h r − , i ( r,n ) , which proves the first part of the claim.For the second part of the claim, we just want to see that for i = 1 , , . . . , r − B i,rn − − X b = B i, r + r ··· + rn − z ω ( b ) + B i +1 , r + r ··· + rn − X b = B i +1 , z ω ( b ) = zf h r − ,r − ℓ − i ( r,n ) . However, by the inductive hypothesis and equation (3) it follows that B i,rn − − X b = B i, r + r ··· + rn − z ω ( b ) + B i +1 , r + r ··· + rn − X b = B i +1 , z ω ( b ) = r − X ℓ = i f h r − ,ℓ i ( r,n − + z i X ℓ =0 f h r − ,ℓ i ( r,n − , = f h r − ,r − ℓ − i ( r,n ) . Thus, the claim holds for all r ≥ n ≥
1. The desired expression for h ∗ ( B ( r,n ) ; z ) then follows immediately from [4, Theorem 2.5]. (cid:3) Example 4.2 (The h ∗ -polynomial of a base-4 simplex) . Example 4.1 presentsthe h ∗ -polynomial of the base-4 3-simplex B (4 , in terms of the average weightedheight statistic. We can recompute this polynomial using the formula proved in IMPLICES FOR NUMERAL SYSTEMS 15
Theorem 4.2. If we expand the polynomial f (4 , and write it diagrammatically as f (4 , = (1 + z + z + z ) , = 1 + 3 z + 6 z + 10 z + 12 z + 12 z + 10 z + 6 z + 3 z + z , = 1 + 10 z · + 10 z · + z · + 3 z · + 12 z · + 6 z · + 6 z · + 12 z · + 3 z · , then we see by the decomposed presentation of f (4 , in the third equality that f h , i (4 , = 1 + 10 z + 10 z + z ,f h , i (4 , = 3 + 12 z + 6 z , and f h , i (4 , = 6 + 12 z + 3 z . Then, by Theorem 4.2, we know that h ∗ ( B (4 , ; z ) = (1 + 10 z + 10 z + z ) + z (9 + 24 z + 9 z ) , = 1 + 19 z + 34 z + 10 z , (4)and thus we recover the h ∗ -polynomial originally computed in Example 4.1. Remark h ∗ -polynomial) . The first line ofequation (4) in Example 4.2 highlights a more general phenomenon. It is a well-known result that if P is a lattice polytope containing an interior lattice point, thenthere is a unique decomposition of h ∗ ( P ; z ) as h ∗ ( P ; z ) = a ( z ) + zb ( z ) , where a ( z ) = z d a (cid:0) z (cid:1) and b ( z ) = z d − b (cid:0) z (cid:1) . Moreover, these polynomials admit anice combinatorial interpretation as a ( z ) = X ∆ ∈ T h (link T (∆); z ) B ∆ ( z ) , and b ( z ) = 1 z X ∆ ∈ T h (link(∆); z ) B conv(∆ , ( z ) , where T is any triangulation of the boundary of P , h (link T (∆); z ) is the h -polynomialof the link of the simplex ∆ in T , and B S ( z ) is the box polynomial of a simplex S . A summary of these various definitions and a proof of this decomposition of h ∗ ( P ; z ) is provided in [1, Chapter 10, Theorem 10.5].For the base- r n -simplex B ( r,n ) , it follows from Theorem 4.2 that a ( z ) = f h r − , i ( r,n ) and b ( z ) = r − X ℓ =1 f h r − ,ℓ i ( r,n ) . In Theorem 4.5, we will use the formulation of h ∗ ( B ( r,n ) ; z ) given in Theorem 4.2to prove that h ∗ ( B ( r,n ) ; z ) is real-rooted and unimodal. In fact, it follows along theway, that the symmetric polynomials a ( z ) and b ( z ) for B ( r,n ) have these propertiesas well. To the best of the author’s knowledge, this is the first known proof ofreal-rootedness of a (non-symmetric) h ∗ -polynomial that comes by way of provingits symmetric decomposition consists of real-rooted polynomials as well. We also note that Theorem 4.2 provides us with a second combinatorial interpre-tation of the coefficients of h ∗ ( B ( r,n ) ; z ). Given a subset S ⊂ Z > and two integers t, m ∈ Z > , we let comp t ( m ; S ) denote the number of compositions of m of length t with parts in S . Since for all k ∈ Z ≥ the coefficient of z k in f ( r,n ) is[ z k ] .f ( r,n ) = comp n ( n + k ; [ r ]) , then we have the following corollary to Theorem 4.2. Corollary 4.3.
For integers r ≥ and n ≥ z k ] .h ∗ ( B ( r,n ) ; z ) = comp n ( n + k ; [ r ]) + r − X ℓ =1 comp n ( n + ( k − r −
1) + ℓ ; [ r ]) , for each k = 0 , , . . . , n . We now use Theorem 4.2 to verify that h ∗ ( B ( r,n ) ; z ) is real-rooted. To do so, wefirst prove that a useful polynomial map G preserves (strict) interlacing. Lemma 4.4.
The polynomial map G := z + 1 1 1 · · · z z + 1 1 ... z z z + 1 . . . ... . . . . . . z z · · · z z + 1 ∈ R [ z ] ( r − × ( r − preserves strict interlacing.Proof. In [5, Theorem 7.8.5] Br¨and´en gives a complete characterization of all suchmatrices. Applying this characterization, it suffices to prove that each of the five2 × (cid:18) (cid:19) , (cid:18) z zz z (cid:19) , (cid:18) z + 1 1 z z + 1 (cid:19) , (cid:18) z + 1 1 (cid:19) , and (cid:18) z z + 1 z z (cid:19) preserve interlacing and nonnegativity. This follows from a series of results in [10,Section 3.11] proven by Fisk. In particular, the result follows for the first twomatrices by applying [10, Lemma 3.71], for the third matrix by [10, Lemma 3.79],and for the fourth matrix by [10, Lemma 3.83(1)]. Finally, the fifth matrix is seento preserve interlacing and nonnegativity by factoring it as (cid:18) z z + 1 z z (cid:19) = (cid:18) (cid:19) (cid:18) z z (cid:19) , and applying [10, Lemma 3.71] to each of these factors. (cid:3) Using Lemma 4.4, we can now prove our main result of this subsection.
Theorem 4.5.
For two integers r ≥ and n ≥ , the h ∗ -polynomial of the base- rn -simplex B ( r,n ) is real-rooted and thus unimodal.Proof. By Theorem 4.2 we know that the h ∗ -polynomial of B ( r,n ) is expressible as h ∗ ( B ( r,n ) ; z ) = f h r − , i ( r,n ) + z r − X ℓ =1 f h r − ,ℓ i ( r,n ) . IMPLICES FOR NUMERAL SYSTEMS 17
Notice next that f ( r,n ) = (1 + z + z + · · · + z r − ) f ( r,n − . (5)For each index k , let a k := [ z k ] .f ( r,n − and b k := [ z k ] .f ( r,n ) . Then recall that wecan write each k uniquely as k = i ( r −
1) + j for integers i and 0 ≤ j < r −
1. Itfollows that[ z i ] .f h r − ,j i ( r,n − = a i ( r − j , and similarly [ z i ] .f h r − ,j i ( r,n ) = b i ( r − j . Therefore, for each ℓ = 0 , , . . . , r −
2, it follows from equation (5) that[ z i ] .f h r − ,ℓ i ( r,n ) = b i ( r − ℓ = ℓ X j =0 a i ( r − ℓ − j ) + r − X j = ℓ a ( i − r − r − ℓ − j ) . For an example of this computation, we refer the reader to Example 4.2. Moreconcisely, this expression for [ z i ] .f h r − ,ℓ i ( r,n ) for each index i is equivalent to sayingthat the vector of polynomials (cid:16) f h r − ,r − i ( r,n ) · · · f h r − , i ( r,n ) f h r − , i ( r,n ) (cid:17) T is produced by multiplying the vector of polynomials (cid:16) f h r − ,r − i ( r,n − · · · f h r − , i ( r,n − f h r − , i ( r,n − (cid:17) T on the left by the ( r − × ( r −
1) matrix of polynomials G in Lemma 4.4. Thus, byway of induction, Lemma 4.4 implies that f h r − ,r − i ( r,n ) ≺ · · · ≺ f h r − , i ( r,n ) ≺ f h r − , i ( r,n ) isa sequence of strictly mutually interlacing and nonnegative polynomials. Moreover,it is well-known that the polynomial map H := · · · z z z z z · · · z ∈ R [ z ] ( r − × ( r − also preserves interlacing. For instance, proofs of this fact can be found in [10,Example 3.73], [5, Corollary 7.8.7], and [14, Proposition 2.2]. Applying this maponce to the vector of polynomials (cid:16) f h r − ,r − i ( r,n ) · · · f h r − , i ( r,n ) f h r − , i ( r,n ) (cid:17) T produces a vector of polynomials (cid:0) g r − · · · g g , (cid:1) T and it follows that g r − ≺ · · · ≺ g ≺ g is a sequence of strictly mutually interlacingpolynomials with the property that g = h ∗ ( B ( r,n ) ; z ). The result then follows. (cid:3) Remark . In order to prove Theorem 4.5 we used Lemma 4.4 to first show that (cid:16) f h r − ,r − ℓ − i ( r,n ) (cid:17) r − ℓ =18 LIAM SOLUS is a strictly interlacing sequence. Other important h -polynomials have been shownto be real-rooted using a closely related construction. In particular, in order toverify a conjecture of [2], Jochemko shows in [14] that the sequence (cid:16) f h r,r − ℓ i ( r,n ) (cid:17) rℓ =1 is strictly interlacing. Similarly, in [16] and [20] Leander and Zhang independentlyshowed that (cid:16) f h r +1 ,r − ℓ +1 i ( r,n ) (cid:17) r +1 ℓ =1 is a strictly interlacing sequence in order to prove that the r th edgewise and clustersubdivisions of the simplex have real-rooted local h -polynomials. Each of thesestrictly interlacing sequences constitutes a distinct family of real-rooted polynomi-als, and collectively they represent the growing prevalence of decompositions of thepolynomial f ( r,n ) in unimodality questions for h -polynomials.5. A Closing Remark
To conclude our discussions, we remark that two natural classes of simplicesassociated to numeral systems have been introduced and analyzed in this note. InSection 3, we searched for numeral systems admitting divisors systems that allowedus to construct a q -vector yielding a reflexive n -simplex with normalized volume the n th place value for all n ≥
0. In the identified examples, we saw that these numeralsystems yield combinatorial interpretations of the associated h ∗ -polynomials thatare closely related to interpretations for the associated q -vectors. Furthermore,these examples all exhibited the desirable distributional properties implied by real-rootedness. To produce more examples of this nature, the (seemingly difficult)problem is to identify a divisor system for some numeral system a , and then studythe geometry of the simplex whose q -vector is given by identity (2) in Definition 3.1.On the other hand, in Section 4, we used a natural choice of q -vector for anymixed radix numeral system to generalize the geometry associated to the binarynumeral system in Theorem 3.6. The result is a family of nonreflexive simplicesassociated to the base- r numeral systems for r ≥
2. We observed that while symme-try of the h ∗ -polynomial is lost in this generalization, real-rootedness is preserved.This suggests a possible extension to a larger family of simplices with real-rooted h ∗ -polynomials, namely, the simplices associated to mixed radix systems via ananalogous choice of q -vector. Amongst such mixed radix simplices, the simplices B ( r,n ) correspond exactly to mixed radix systems in which all radices are taken tobe equal, and our proof of real-rootedness relies heavily on this fact. Thus, newtechniques may be necessary to address real-rootedness for this larger family. On arelated note, computations suggest that the simplices B ( r,n ) are also Ehrhart posi-tive . This curious observation further serves to promote the study of simplices fornumeral systems from the combinatorial perspective.
Acknowledgements . The author was supported by an NSF Mathematical Sci-ences Postdoctoral Research Fellowship (DMS - 1606407). He would also like tothank Petter Br¨and´en and Benjamin Braun for helpful discussions on the project.
IMPLICES FOR NUMERAL SYSTEMS 19
References [1] M. Beck and S. Robins.
Computing the continuous discretely.
Springer Science+ BusinessMedia, LLC, 2007.[2] M. Beck and A. Stapledon.
On the log-concavity of Hilbert series of Veronese subrings andEhrhart series.
Mathematische Zeitschrift 264.1 (2010): 195-207.[3] B. Braun and R. Davis.
Ehrhart series, unimodality, and integrally closed reflexive polytopes.
Annals of Combinatorics 20.4 (2016): 705-717.[4] B. Braun, R. Davis, and L. Solus.
Detecting the integer decomposition property and Ehrhartunimodality in reflexive simplices.
Submitted to Discrete and Computational Geometry.Preprint available at https://arxiv.org/abs/1608.01614 (2016).[5] P. Br¨and´en.
Unimodality, log-concavity, real-rootedness and beyond.
Handbook of Enumera-tive Combinatorics (2015): 437-483.[6] P. Candelas, C. Xenia, P. S. Green, and L. Parkes.
A pair of Calabi-Yau manifolds as anexactly soluble superconformal theory.
Nuclear Physics B 359.1 (1991): 21-74.[7] H. Conrads.
Weighted projective spaces and reflexive simplices. manuscripta mathematica107.2 (2002): 215-227.[8] D. A. Cox and S. Katz.
Mirror symmetry and algebraic geometry.
No. 68. American Mathe-matical Soc., 1999.[9] E. Ehrhart.
Sur les polyh`edres rationnels homoth`etiques `a n dimensions . C. R. Acad Sci.Paris, 254:616-618, 1962.[10] S. Fisk. Polynomials, roots, and interlacing.
Preprint available at https://arxiv.org/abs/math/0612833 (2008).[11] A. S. Fraenkel.
Systems of numeration.
The American Mathematical Monthly, Vol. 92, No.2 (Feb., 1985), pp. 105-114.[12] T. Hibi.
Note dual polytopes of rational convex polytopes.
Combinatorica 12.2 (1992).[13] T. Hibi, M. Olsen, and A. Tsuchiya.
Self-dual reflexive simplices with Eulerian polynomials.
Graphs and Combinatorics DOI 10.1007/s00373-017-1781-8, 2017.[14] K. Jochemko.
On the real-rootedness of the Veronese construction for rational formal powerseries.
Preprint available at https://arxiv.org/abs/1602.09139 (2016).[15] D. E. Knuth.
The art of computer programming: sorting and searching.
Vol. 3. PearsonEducation, 1998.[16] M. Leander.
Compatible polynomials and edgewise subdivisions.
Preprint available at https://arxiv.org/abs/1605.05287 (2016).[17] B. Nill.
Volume and lattice points of reflexive simplices.
Discrete & Computational Geometry37.2 (2007): 301-320.[18] N. J. Sloane.
The On-Line Encyclopedia of Integer Sequences . (2003).[19] R. P. Stanley.
Decompositions of rational convex polytopes.
Annals of discrete mathematics6 (1980): 333-342.[20] P. B. Zhang.
On the Real-rootedness of the Local h -polynomials of Edgewise Subdivisions ofSimplexes. Preprint available at https://arxiv.org/abs/1605.02298 (2016).
Matematik, KTH, SE-100 44 Stockholm, Sweden
E-mail address ::