VARGA S THEOREM IN NUMBER FIELDS
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1 VARGA S THEOREM IN NUMBER FIELDS PETE L. CLARK AND LORI D. WATSON Abstract. We give a number field version of a recent result of Varga on solutions of polynomial equations with binary input variables and relaxed output variables. 1. Introduction This note gives a contribution to the study of solution sets of systems of polynomial equations over finite local principal rings in the restricted input / relaxed output setting. The following recent result should help to explain the setting and scope. Let n, a 1,..., a n Z + and 1 N n i=1 a i. Put { 1 if N < n ma 1,..., a n ; N = min n i=1 y i if n N n i=1 a ; i the minimum is over y 1,..., y n Z n with 1 y i a i for all i and n i=1 y i = N. Theorem 1.1. [Cl18, Thm. 1.7] Let R be a Dedekind domain, and let p be a maximal ideal in R with finite residue field R/p = F q. Let n, r, v 1,..., v r Z +. Let A 1,..., A n, B 1,..., B r R be nonempty subsets each having the property that no two distinct elements are congruent modulo p. Let r, v 1,..., v r Z +. Let P 1,..., P r R[t 1,..., t n ] be nonzero polynomials, and put n za B := #{x A i 1 j m P j x B j mod p vj }. i=1 Then za B = 0 or n za B m #A 1,..., #A n ; #A i i=1 r q vj #B j degp j. Remark 1.2. For every finite local principal ring r, there is a number field K, a prime ideal p of the ring of integers Z K of K, and v Z + such that r = Z K /p v [Ne71], [BC15]. Henceforth we will work in the setting of residue rings of Z K. If in Theorem 1.1 we take v 1 = = v r = 1, A i = F q for all i and B j = {0} for all j, then we recover a result of E. Warning. Theorem 1.3. Warning s Second Theorem [Wa35] Let P 1,..., P r F q [t 1,..., t n ] be nonzero polynomials, and let z = #{x = x 1,..., x n F n q P 1 x = = P r x = 0}. Then z = 0 or z q n n degpj. 1
2 2 PETE L. CLARK AND LORI D. WATSON By Remark 1.2, we may write F q as Z K /p for a suitable maximal ideal p in the ring of integers Z K of a suitable number field K. Having done so, Theorem 1.3 can be interpreted in terms of solutions to a congruence modulo p, whereas Theorem 1.1 concerns congruences modulo powers of p. At the same time, we are restricting the input variables x 1,..., x n to lie in certain subsets A 1,..., A n and also relaxing the output variables: we do not require that P j x = 0 but only that P j x lies in a certain subset B j modulo p vj. There is however a tradeoff: Theorem 1.1 contains the hypothesis that no two elements of any A i resp. B j are congruent modulo p. Thus, whereas when v j = 1 for all j we are restricting variables by choice e.g. we could take each A i to be a complete set of coset representatives for p in Z K as done above when v j > 1 we are restricting variables by necessity we cannot take A i to be a complete set of coset representatives for p vj in Z K. We would like to have a version of Theorem 1.1 in which the A i s can be any nonempty finite subsets of Z K and the B j can be any nonempty finite subsets of Z K containing {0}. However, to do so the degree conditions need to be modified in order to take care of the arithmetic of the rings Z K /p dj. In general this seems like a difficult and worthy problem. An interesting special case was resolved in recent work of L. Varga [Va14]. His degree bound comes in terms of a new invariant of a subset B Z/p d Z \ {0} called the price of B and denoted prb that makes interesting connections to the theory of integer-valued polynomials. Theorem 1.4. Varga [Va14, Thm. 6] Let P 1,..., P r Z[t 1,..., t n ] \ {0} be polynomials without constant terms. For 1 j r, let d j Z +, and let B j Z/p dj Z be a subset containing 0. If r degp j prz/p dj Z \ B j < n, then #{x {0, 1} n 1 j r, P j x B j mod p dj } 2. In this note we will revisit and extend Varga s work. Here is our main result. Theorem 1.5. Let K be a number field of degree N, and let e 1,..., e N be a Z-basis for Z K. Let p be a nonzero prime ideal of Z K, and let d 1,..., d r Z +. Let P 1,..., P r Z K [t 1,..., t n ] be nonzero polynomials without constant terms. For each 1 j r, there are unique {ϕ j,k } 1 k N Z[t 1,..., t n ] such that N 1 P j t = ϕ j,k e j. For 1 j r, let B j be a subset of Z K /p dj that contains 0 mod p dj. Let r N S := degϕ j,k prz K /p dj \ B j. Then #{x {0, 1} n 1 j r, P j x mod p dj B j } 2 n S. Thus we extend Varga s Theorem 1.4 from Z to Z K and refine the bound on the number of solutions. In 2 we discuss the price of a subset of Z K /p d. It seems to us that Varga s definition of the price has minor technical flaws: as we understand it, he tacitly assumes that for an integer-valued polynomial f Q[t] and m, n Z, the output fn modulo m depends only on the input modulo m. This is not true: for instance if ft = tt 1 2, then fn modulo 2 depends on n modulo 4, not
3 VARGA S THEOREM IN NUMBER FIELDS 3 just modulo 2. So we take up the discussion from scratch, in the context of residue rings of Z K. The proof of Theorem 1.5 occupies 3. After setting notation in 3.1 and developing some preliminaries on multivariate Gregory-Newton expansions in 3.2, the proof proper occurs in The Price Consider the ring of integer-valued polynomials We have inclusions of rings Let IntZ K, Z K = {f K[t] fz K Z K }. Z K [t] IntZ K, Z K K[t]. mp, 0 := {f IntZ K, Z K f0 0 mod p}. Observe that mp, 0 is the kernel of a ring homomorphism IntZ K, Z K Z K /p: first evaluate f at 0 and then reduce modulo p. So mp, 0 is a maximal ideal of IntZ K, Z K. We put Up, 0 := IntZ K, Z K \ mp, 0 = {f IntZ K, Z K f0 / p}. Let d Z +, and let B be a subset of Z K /p d. We say that h Up, 0 covers B if: for all b Z K such that b mod p d B, we have hb p. The price of B, denoted prb, is the least degree of a polynomial h Up, 0 that covers B, or if there is no such polynomial. Remark 2.1. a If B 1, B 2 are subsets of Z K /p d \ {0}, then prb 1 B 2 prb 1 + prb 2 : If for i = 1, 2 the polynomial h i Up, 0 covers B i and has degree d i, then h 1 h 2 Up, 0 covers B 1 B 2 and has degree d 1 + d 2. b If 0 mod p d B, then prb = : Since 0 B we need h0 p, contradicting h Up, 0. c If d = 1, then for any subset B Z K /p \ {0}, we have prb #B: Let B be any lift of B to Z K. Then h = x Bt x Z K [t] IntZ K, Z K covers B and has degree #B. Note that here we use polynoimals with Z K -coefficients. It is clear that #B is the minimal degree of a covering polynomial h with Z K -coefficients: we can then reduce modulo p to get a polynomial in F q [t] that we want to be 0 at the points of B and nonzero at 0, so of course it must have degree at least #B. d If we assume no element of B is 0 modulo p, let B be the image of B under the natural map Z K /p d Z K /p = F q ; then our assumption gives 0 / B. Above we constructed a polynomial h Z K [t] of degree #B such that h0 / p and for all x Z K such that x mod p B, we have hx p. This same polynomial h covers B and shows that prb prb #B. For B Z K /p d \ {0} we define κb Z +, as follows. For 1 i d we will recursively define B i Z K /p i \ {0} and k i 1 N. Put B d = B, and let k d 1 be the number of elements of B d that lie in p d 1. Having defined B i and k i 1, we let B i 1 be the set of x Z K /p i 1 such that there are more than k i 1 elements of B i mapping to x under reduction modulo p i 1. We let k i 2 be the number of elements of B i 1 that lie in p i 2.
4 4 PETE L. CLARK AND LORI D. WATSON Notice that 0 / B i for all i: indeed, B i is defined as the set of elements x such that the fiber under the map Z K /p i+1 Z K /p i has more elements of B i+1 than does the fiber over 0. We put Lemma 2.2. We have κb q d 1. d 1 κb := k i q i. Proof. Each k i is a set of elements in a fiber of a q-to-1 map, so certainly k i q. In order to have k i = q, then B i+1 would need to contain the entire fiber over 0 Z K /p i, but this fiber includes 0 Z K /p i+1, which as above does not lie in B i+1. So i=0 d 1 d 1 κb = k i q i q 1q i = q d 1. i=0 Theorem 2.3. For any subset B Z K /p d \ {0}, we have prb κb. i=0 Proof. Step 1: For r 1, let A = {a 1,..., a q d 1} Z K /p d be a complete residue system modulo p d 1 none of whose elements lie in p d. We will show how to cover A with f Up, 0 of degree q d 1. We denote by v p the p-adic valuation on K. Let λ Z K be an element with v p λ = d 2 j=0 qj, and let β Z K be an element such that v p β = 0 and for all nonzero prime ideals q p of Z K, we have v q β v q λ. Such elements exist by the Chinese Remainder Theorem. Put g A t := t a j Z K [t], h A t := β λ g At K[t]. q d 1 For all x Z K, {x a 1,..., x a q d 1} is a complete residue system modulo p d 1, so in q d 1 x a j, for all 0 j d 1 there are q d 1 j factors in p j, so v p g A x d 2 j=0 qj and thus v p h A x 0. For any prime ideal q p of Z K, both v q g A x and v q β λ are non-negative, so v q h A x 0. Thus h A Int Z K. Moreover the condition that no a j lies in p d ensures that v p g A 0 = d 2 j=0 qj, so h A Up, 0. If x Z K is such that x a j mod p d for some j, then v p x a j d. Since in the above lower bounds of v p g A x we obtained a lower bound of at most d 1 on the p-adic valuation of each factor, this gives an extra divisibility and shows that v p h A x 0. Thus h A covers A with price at most q d 1. Step 2: Now let B Z K /p d \ {0}. The number of elements of B that lie in p d 1 is k d 1. For each of these elements x i we choose a complete residue system A i modulo p d 1 containing it; since no x i lies in p d this system satisfies the hypothesis of Step 1, so we can cover each A i with price at most q d 1 and thus using Remark 2.1a all of the A i s with price at most k d 1 q d 1. However, by suitably choosing the A i s we can cover many other elements as well. Indeed, because we are choosing k d 1 complete residue systems modulo p d 1, we can cover every element x that is congruent modulo p d 1 to at most k d 1 elements of B. By definition of B d 1, this means that we can cover all elements of B that do not map modulo p d 1 into B d 1. Now suppose that we can cover B d 1 by h Up, 0 of degree κ. This means that for every x Z K such that x mod p d 1 lies in B d 1, hx p. But then every element of B whose image in p d 1 lies in B d 1 is covered by h, so altogether we get prb k d 1 q d 1 + prb d 1. Now applying the same argument successively to B d 1,..., B 1 gives prb i k i 1 q i 1 + prb i 1
5 VARGA S THEOREM IN NUMBER FIELDS 5 and thus d 1 prb k i q i = κb. i=0 3. Proof of the Main Theorem 3.1. Notation. Let K be a number field of degree N, and let e 1,..., e N be a Z-basis for Z K. A Z-basis for Z K [t 1,..., t n ] is given by e j t I as j ranges over elements of {1,..., N} and I ranges over elements of N n. So for any f Z K [t 1,..., t n ], we may write 2 f = ϕ 1 t 1,..., t n e ϕ N t 1,..., t n e N, ϕ i Z[t 1,..., t n ]. Then we have For a subset B Z K /p d, we put 3.2. Multivariable Newton Expansions. deg f = max deg ϕ i. i B = Z K /p d \ B. Lemma 3.1. If f Q[t] is a polynomial and fn Z, then fz Z. Proof. See e.g. [CC, p. 2]. Theorem 3.2. Let f K[t]. a There is a unique function α f : N N K, r α r f such that i we have α r f = 0 for all but finitely many r N N, and ii for all x = x 1 e x N e N Z K, we have 3 fx = r N N α r f x1 r 1 b The following are equivalent: i We have f IntZ K, Z K. ii For all r N N, α r f Z K. We call the α r f the Gregory-Newton coefficients of f. Proof. Step 1: Let f K[t]. Let e 1,..., e N be a Z-basis for Z K. We introduce new independent indeterminates t 1,..., t N and make the substitution N t = e k t k to get a polynomial f K[t]. This polynomial induces a map K N K hence, by restriction, a map Z N K. For x = x 1,..., x N Z N, write x = x 1 e x N e N Z K. Then we have fx = fx. Let M = MapsZ N, K be the set of all such functions, and let P be the K-subspace of M consisting of functions obtained by evaluating a polynomial in K[t] on Z N, as above. By the CATS Lemma [Cl14, Thm. 12], the map K[t] P is an isomorphism of K-vector spaces. Henceforth we will identify K[t] with P inside M. xn r N.
6 6 PETE L. CLARK AND LORI D. WATSON Step 2: For all 1 k N, we define a K-linear endomorphism k of M, the kth partial difference operator: k g : x Z N gx + e k gx. These endomorphisms all commute with each other: i j g = gx + e i + e j gx + e j gx + e i + gx = j i g. Let 0 k be the identity operator on M, and for i Z+, let i k be the i-fold composition of k. For I = i 1,..., i N N N, put I = i1... i N End K M. When we apply k to a monomial t I, we get another polynomial. More precisely, if deg tk t I = 0 then k t I is the zero polynomial; otherwise deg tk k t I = deg tk t I 1; l k, deg tl k t I = deg tl t I. Thus for each f P, for all but finitely many I N N, we have that I f = 0. For the one variable difference operator, we have x x + 1 x x = =. r r r r 1 From this it follows that for I, r N N we have 4 I x1 xn = = δ r,i r 1 r 1 i 1 r N i N r N So if β : N N K is any finitely nonzero function then for all I N N we have 5 I r N N β r x1 r 1 xn r N 0 = β I, and thus there is at most one such function satisfying 3, namely α f : r r f0. So for any f M and r N N, we define the Gregory-Newton coefficient α r f := r f0 K. We may view the assignment of the package {α r f} r N N of Gregory-Newton coefficients to f M as a K-linear mapping M K NN. If we put M + = MapsN N, K, then we get a factorization M M + α K N N, where the first map restricts from Z N to N N, and the factorization occurs because the Gregory- Newton coefficients depend only on the values of f on N N. We make several observations: First Observation: The map α is an isomorphism. Indeed, knowing all the successive differences at 0 is equivalent to knowing all the values on N N, and all possible packages of Gregory-Newton coefficients arise. Namely, let S n be the assertion that for all x N N with k x k = n and all f M, then fx is a Z-linear combination of its Gregory-Newton coefficients. The case n = 0 is clear: f0 = α 0 f. Suppose S n holds for n, let x N N be such that k x k = n + 1, and choose k such that x = y + e k ; thus k y k = n. Then fx = fy + k fy.
7 VARGA S THEOREM IN NUMBER FIELDS 7 By induction, fy is a Z-linear combination of the Gregory-Newton coefficients of f and k fy is a Z-linear combination of the Gregory-Newton coefficients of k f. But every Gregory-Newton coefficient of k f is also a Gregory-Newton coefficient of f, completing the induction. Second Observation: The composite map K[t] M M + α K N N is an injection. Indeed, the kernel of M K NN is the set of functions that vanish on Z N \ N N. In particular, any element of the kernel vanishes on the infinite Cartesian subset Z <0 N and thus by the CATS Lemma is the zero polynomial. Third Observation: For a subring R K and f M, we have fn N R iff all of the Gregory-Newton coefficients of f lie in R. This is a consequence of the First Observation: the Gregory-Newton coefficients are Z-linear combinations of the values of f on N N and conversely. Step 3: For F K[t], we define the Newton expansion T F = t1 tn α r F K[t]. r 1 r N r N N This is a finite sum. Moreover, by definition of α r F and by 5 we get that for all r N N, α r T F = α r F. It now follows from Step 2 that T F = F K[t]. Applying this to the f associated to f K[t] in Step 1 completes the proof of part a. Step 4: If we assume that f IntZ K, Z K then fz N Z K so all the Gregory-Newton coefficents lie in Z K. Conversely, if all the Gregory-Newton coefficients of f lie in Z K, then for x = x 1 e x N e N Z K, by Lemma 3.1 and 3 we have fx = fx 1,..., x N Z K, so f IntZ K, Z K Proof of Theorem 1.5. We begin by recalling the following result. Theorem 3.3. Let F be a field, and let P F [t 1,..., t n ] be a polynomial. Let Then either #U = 0 or #U 2 n degp. U := {x {0, 1} n P x 0}. Proof. This is a special case of a result of Alon-Füredi [AF93, Thm. 5]. We now turn to the proof of Theorem 1.5. Put Z := {x {0, 1} n 1 j r, P j x mod p dj B j }. Step 0: If q = #Z K /p is a power of p, then we have p d p d. Therefore in 1 if we modify any coefficient of ϕ j,k t by a multiple of p d, it does not change P j modulo p d and thus does not change the set Z. We may thus assume that every coefficient of every ϕ j,k is non-negative. Step 1: For w = k i=1 tii a sum of monomials and 0 r k, we put Ψ r w := t Ii 1 t I ir. 1 i 1<i 2<...<i r k For x {0, 1} n, we have wx = #{1 i k x Ii = 1}, so wx Ψ r wx =. r For f Z K [t 1,..., t n ], write f = N ϕ kte k and suppose that all the coefficients of each ϕ k are non-negative equivalently, each ϕ k t is a sum of monomials. For r N N, we put Ψ r f := Ψ r1 ϕ 1 Ψ rn ϕ N Z[t].
8 8 PETE L. CLARK AND LORI D. WATSON For h IntZ K, Z K with Gregory-Newton coefficients α r, we put Ψ h f := r N N α r Ψ r f Z K [t]. For x {0, 1} n, we have so using 2 we get Ψ h fx = r Ψ r x = N ϕk x, α r Ψ r fx = r r k ϕ1 x ϕn x α r r 1 r N = hϕ 1 xe ϕ N xe N = hfx. Step 2: For 1 j r, let h j IntZ K, Z K have degree prb j and cover B j. Put r F := Ψ hj P j mod p Z K /p[t] = F q [t]. Note that degf r deg Ψ hj P j r n degh j degϕ j,k = S. Here is the key observation: for x {0, 1} n, if F x 0, then for all 1 j r we have p Ψ hj P j x = h j P j x, so P j x mod p dj / B j, and thus x Z. Step 3: For all 1 j r we have P j 0 = 0 and h j Up, 0, so h j 0 / p, so r r r F 0 = Ψ h j P j 0 = h j P j 0 mod p = h j 0 mod p 0. Applying Alon-Füredi to F, we get completing the proof of Theorem 1.5. #Z #{x {0, 1} n F x 0} 2 n deg F 2 n S, References [AF93] N. Alon and Z. Füredi, Covering the cube by affine hyperplanes. Eur. J. Comb , [BC15] A. Brunyate and P.L. Clark, Extending the Zolotarev-Frobenius approach to quadratic reciprocity. Ramanujan J , [CC] P.-J. Cahen and J.-L. Chabert, Integer-valued polynomials. Mathematical Surveys and Monographs, 48. American Mathematical Society, Providence, RI, [CFS14] P.L. Clark, A. Forrow and J.R. Schmitt, Warning s Second Theorem with restricted variables. Combinatorica , [Cl14] P.L. Clark, The Combinatorial Nullstellensätze Revisited. Electronic Journal of Combinatorics. Volume 21, Issue Paper #P4.15. [Cl18] P.L. Clark, Fattening up Warning s Second Theorem. pdf [Ne71] A.A. Nečaev, The structure of finite commutative rings with unity. Mat. Zametki , [Va14] L. Varga, Combinatorial Nullstellensatz modulo prime powers and the parity argument. Electron. J. Combin , no. 4, Paper 4.44, 17 pp. [Wa35] E. Warning, Bemerkung zur vorstehenden Arbeit von Herrn Chevalley. Abh. Math. Sem. Hamburg ,
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