Hitting Sets

Definition 2.6.6. Let A ⊆ Rⁿ≥0 be a subset and let w ∈ Nⁿ be a weight vector.

(a) Letα∈A. If|α|_w <|β|_w for allβ ∈A\{α}, then we say thatwisolates α in A.

(b) If there exists α ∈ A such that w isolates α in A, then w is called isolating for A.

Let d ≥0 and let f ∈ K[x] be a non-zero polynomial of degree at most d. Then the logarithmic support A := LSupp(f) ⊂ Nⁿ is non-empty. If a weight vector w∈Nⁿ isolates some α∈A, then the univariate polynomial

f z^w1, . . . , z^wn

∈K[z],

is non-zero, because it has a non-zero monomial of degree|α|_w. Note that the Kronecker substitution (2.6.1) yields a weight vector w := 1, D, . . . , Dⁿ⁻¹ for A, though, with entries exponential in d. We are interested in weights of magnitude poly(n, d). The following lemma demonstrates that a weight vector which is randomly chosen from [2nd]ⁿ is isolating for A with high probability.

Lemma 2.6.7 (Isolating Lemma, [KS01, Lemma 4]). Let d, N ≥1, and let A⊂Nⁿ such that |α| ≤d for all α∈A. Then we have

Pr

w∈[N]ⁿ

w is isolating for A

≥1−nd N .

A suitable derandomization of the Isolating Lemma (see for example [AM08]) would imply a deterministic polynomial-time identity test for arith-metic circuits of polynomial degree.

Finally, we remark that it is easy to obtain an isolating weight vector for A ⊂ Nⁿ if the convex polytope Conv(A) has few vertices. We will exploit this fact in Section 3.2.3.

given arithmetic circuit. Algorithms of this kind are referred to as blackbox algorithms. Blackbox algorithms require the computation of a hitting set according to the following definition.

Definition 2.7.1. Let C ⊆K[x] be a set of polynomials. A set H ⊆Kⁿ is called a hitting set for C if for all non-zero C ∈ C there exists a ∈ H such that C(a)6= 0.

Example 2.7.2. Let us give two examples of hitting sets.

(a) Letd≥0 and letS ⊆K be a subset such that|S| ≥d+ 1. ThenSⁿ is a hitting set forK[x]_≤dby Theorem 2.5.4. The size of this hitting set is ex-ponential in the general setting. However, for polynomial-degree circuits with constantly many variables, we obtain a polynomial-size hitting set.

(b) LetK =Qand letC_n,s be the set of arithmetic circuitsC overQ[x] such that size(C) ≤ s. Then the proof of Theorem 2.6.3 yields a hitting set for C_n,s consisting of a single point. The coordinates of this point have bit-size exponential in s.

Existence of small hitting sets

The existence of small hitting sets was proven by Heintz & Schnorr [HS80a]

(in their paper, hitting sets are called “correct test sequences”) for fields of characteristic zero. Here we reproduce their proof, but replace a result they use from [HS80b] by a simpler argument that works for arbitrary fields. This argument is inspired by the proof of [SY10, Theorem 3.1]. We will require some machinery from algebraic geometry which we cover in Appendix A.4.

Theorem 2.7.3. Let 1 ≤ n ≤ s and let d ≥ 1. Let K be a field and let S ⊆K be an arbitrary subset with |S| ≥(2sd+ 2)². Denote by C_n,d,s the set of arithmetic circuits C over K[x]such that fdeg(C)≤d and |C| ≤s. Then there exists a hitting set H_n,d,s ⊆Sⁿ for C_n,d,s such that |H_n,d,s| ≤9s.

Proof. Let y = {y₁, . . . , y_s} be new variables and let S_n,d,s be the set of constant-free arithmetic circuits C over K[x,y] such that fdeg(C) ≤ d and

|C| ≤ s. Obviously, every circuit in C_n,d,s can be obtained from a cir-cuit in S_n,d,s by substituting constants for the y-variables. There are at most s^2s connected, directed multigraphs C with V(C) ⊆ [s] and |C| ≤ s, and the vertices of each such multigraph can be labeled by the symbols {+,×, x₁, . . . , x_n, y₁, . . . , y_s} in at most (2s+ 2)^s different ways. Therefore, we have |S_n,d,s| ≤(2s+ 2)^3s.

Set t:= ^n+d_d

and letx^α1, . . .x^αt ∈T(x) be the terms of degree at most d. We identify a polynomial f = Pt

i=1c_i·x^αi ∈ K[x]≤d with its vector of

coefficients (c₁, . . . , c_t)∈K^t, henceC_n,d,s ⊆K^t. LetC ∈ S_n,d,s be a constant-free circuit. WriteC =Pt

i=1c_i·x^αi withc_i ∈K[y]. The coefficientsc_i define a morphism

ϕ_C: K^s →K^t, a7→(c₁(a), . . . , c_t(a))

with deg(ϕ_C) ≤ d. Let Y_C ⊆ K^t be the Zariski closure of ϕ_C(K^s). Since dim(K^s) = s, we have dim(YC) ≤ s. By Lemma 2.7.4 below, we obtain deg_K^t(Y_C)≤d^s. The affine variety

Y_n,d,s:= [

C∈S_n,d,s

Y_C ⊆K^t

containsC_n,d,s and satisfies dim(Y_n,d,s)≤max{dim(Y_C)|C ∈ S_n,d,s} ≤s and deg_K^t(Y_n,d,s)≤ X

C∈S_n,d,s

deg_K^t(Y_C)

≤ |S_n,d,s| ·max

deg_K^t(Y_C)|C ∈ S_n,d,s

≤(2s+ 2)^3sd^s.

Set m := 9s. We want to show that there exists a tuple of points (a₁, . . . ,a_m)∈S^mn such that for all non-zeroC ∈ C_n,d,s there exists i∈[m]

such thatC(a_i)6= 0. This tuple will then constitute a desired hitting set.

Consider the affine variety X :=

(f,a₁, . . . ,a_m)∈K^t+mn|f ∈Y_n,d,s and f(a_i) = 0 for all i∈[m] . Fori∈[t] and (j, k)∈[m]×[n], letz_i and z_j,k be the coordinates ofK^t+mn. Then X is defined by the polynomial equations for Y_n,d,s and

t

X

i=1

z_i·z_j,1^αi,1· · ·z_j,n^αi,n, j ∈[m].

By Theorem A.4.7, we have

deg_K^t+mn(X)≤deg_K^t+mn(Yn,d,s)·(d+ 1)^m ≤(2s+ 2)^3sd^s(d+ 1)^m. Define the projections π₁: K^t+mn K^t and π₂: K^t+mn K^mn to the first t and last mn coordinates, respectively. Let C₁, . . . , C<sub>`</sub> ⊆ K<sup>t+mn</sup> be all irreducible components C ⊆ X such that π1(C) contains a non-zero polynomial, and set C := S`

i=1C_i. Then π₂(C)∩ S^mn contains all tuples (a₁, . . . ,a_m)∈S^mn that do not constitute a hitting set for C_n,d,s.

Let i∈[`] and let f ∈π₁(C_i) such that f 6= 0. Then π₁⁻¹(f) ={f} × V_Kⁿ(f)× · · · × V_Kⁿ(f)

| {z }

mtimes

,

hence dim(π₁⁻¹(f)) = m(n−1). Applying Lemma A.4.2 to the morphism π₁: C_i →Y_n,d,s, we obtain

dim(C_i)≤dim(π⁻¹₁ (f)) + dim(Y_n,d,s)≤m(n−1) +s.

This implies dim(C)≤max{dim(C_i)|i∈[`]} ≤m(n−1) +s.

Now define the hypersurfaces

Hj,k :=V_K^t+mn Q

c∈S(zj,k−c) for all j ∈[m] andk ∈[n], and set H :=T

(j,k)∈[m]×[n]H_j,k. Then we have

|π2(C)∩S^mn|=|π2(C∩H)|

≤deg_K^t+mn(C∩H)

≤deg_K^t+mn(C)·max{deg_K^t+mn(H_j,k)|j ∈[m], k ∈[n]}^dim(C)

≤deg_K^t+mn(X)· |S|^dim(C)

≤(2s+ 2)^3sd^s(d+ 1)^m· |S|^m(n−1)+s

≤ |S|^2s+m/2· |S|^m(n−1)+s

=|S|^−m/6· |S|^mn,

where the second inequality follows from Corollary A.4.8. This implies the existence of a tuple (a₁, . . . ,a_m) ∈ S^mn that constitutes a hitting set for C_n,d,s.

In the proof of Theorem 2.7.3 we used the following lemma for bounding the degree of the image of a morphism.

Lemma 2.7.4. Let X ⊆ K^s be an irreducible affine variety, let Y ⊆K^t be an affine variety, and letϕ:X →Y be a dominant morphism. Then we have

deg_K^t(Y)≤deg_K^s(X)·deg(ϕ)^dim(Y).

Proof. The following argument is contained in the proof of [HS80b, Lemma 1]. Set r := dim(Y). Since ϕ is dominant, Y is irreducible. By Theorem A.4.4, ϕ(X) is a constructible set, so by Lemma A.4.3 it contains a non-empty open subset of Y. Therefore, by Lemma A.4.6, there exist affine hyperplanes H₁, . . . , H_r⊂K^t such that deg_K^t(Y) = |ϕ(X)∩H₁∩ · · · ∩H_r|.

Thenϕ⁻¹(H_i)⊂K^s is an affine hypersurface with deg_K^s(ϕ⁻¹(H_i))≤deg(ϕ) for all i∈[r]. By Theorem A.4.7, we obtain

deg_K^s X∩ϕ⁻¹(H₁)∩ · · · ∩ϕ⁻¹(H_r)

≤deg_K^s(X)·deg(ϕ)^r.

Let C₁, . . . , C_m ⊆ K^s be the irreducible components of the affine variety X∩ϕ⁻¹(H₁)∩ · · · ∩ϕ⁻¹(H_r). Since the map

ϕ: X∩ϕ⁻¹(H₁)∩ · · · ∩ϕ⁻¹(H_r)→ϕ(X)∩H₁∩ · · · ∩H_r is surjective andϕ(C_i) is a singleton for all i∈[m], we get

deg_K^t(Y) =|ϕ(X)∩H₁∩ · · · ∩H_r|

≤m

≤

m

X

i=1

deg_K^s(C_i)

= deg_K^s X∩ϕ⁻¹(H₁)∩ · · · ∩ϕ⁻¹(H_r)

≤deg_K^s(X)·deg(ϕ)^r, finishing the proof.

Polynomial-space computation of hitting sets

Using quantifier elimination, the proof of Theorem 2.7.3 can be turned into a polynomial-space algorithm for the computation of small hitting sets. For an introduction to quantifier elimination, see [BPR06, Chapter 1].

Theorem 2.7.5. Let K = Q or K = F^q for some prime power q. Then there exists a Turing machine that, given 1≤n ≤s and d ≥1, computes in poly(s)-space a hitting set H_n,d,s ⊆ Sⁿ for C_n,d,s of size |H_n,d,s| ≤ 9s, where S ⊂K is a subset such that bs(c) = poly(logs,logd) for all c∈S.

Proof sketch. The description of the asserted Turing machineM is as follows.

If K = Q, then M sets S ← [(2sd + 2)²] ⊂ K. If K = F^q for some prime power q, then M constructs the smallest field extension L/K such that |L| ≥ (2sd + 2)² and picks a subset S ⊆ L of size |S| = (2sd+ 2)². In both cases, we obtain a subset S ⊆ K such that |S| = (2sd + 2)² and bs(c) = poly(logs,logd) for all c∈S.

Next, M sets m ← 9s and checks for all m-subsets H ⊆ Sⁿ whether H is a hitting set for Cn,d,s as follows. As in the proof of Theorem 2.7.3, let y={y₁, . . . , y_s}be new variables and letS_n,d,s be the set of all constant-free arithmetic circuits C over K[x,y] such that fdeg(C) ≤ d and |C| ≤ s. Let

C_n,d,s be the set of arithmetic circuits C over K[x] such that fdeg(C) ≤ d and |C| ≤s, thusC_n,d,s ⊆ C_n,d,s. ThenH is a hitting set forC_n,d,s if and only if the sentence

^

C∈S_n,d,s

∀c∈K^s

∀b∈Kⁿ C(b,c) = 0

∨ _

a∈H

C(a,c)6= 0

(in the first-order theory of algebraically closed fields) is true. Note that the sentence in the innermost parentheses is just another way of saying that C(x,c) ∈ K[x] is the zero polynomial. Using quantifier elimination, M checks the truth of the sentence

∀c∈K^s

∀b∈Kⁿ C(b,c) = 0

∨ _

a∈H

C(a,c)6= 0

for all C ∈ S_n,d,s. Since the number of quantifier alternations is constant, this can be done in poly(s)-space [Ier89].

By Theorem 2.7.3, M will eventually find a hitting set H_n,d,s ⊆ Sⁿ for C_n,d,s of size |H_n,d,s| ≤ 9s. By reusing space, the algorithm can be imple-mented to run in poly(s)-space.

Connections to lower bounds

The following simple theorem demonstrates that small hitting sets imply lower bounds (cf. [HS80a, Theorem 4.5]). See also [Agr05] for a similar result.

Theorem 2.7.6. Let 1≤ n ≤s and let d ≥1. Let K be a field, let S ⊆ K be a subset, and let K₀ ⊆ K be the prime field of K. Denote by C_n,d,s the set of arithmetic circuits C over K[x] such that fdeg(C) ≤ d and |C| ≤ s.

Assume that H_n,d,s ⊆Sⁿ is a hitting set for C_n,d,s of size m :=|H_n,d,s|.

If m≤ ^n+d_d

−1, then there exists a non-zero polynomial f ∈K₀(S)[x]≤d

with sp(f)≤m+ 1 such that f /∈ Cn,d,s.

Proof. The proof is by interpolation. Denote H_n,d,s = {a₁, . . . ,a_m}, let t₁, . . . , t_m+1 ∈ T(x)≤d be distinct terms, and let y = {y₁, . . . , y_m+1} be new variables. Consider the homogeneous system of linear equations

t₁(a_i)·y₁+· · ·+t_m+1(a_i)·y_m+1 = 0, i∈[m],

with indeterminatesy and coefficients inK0(S). Since this system has more variables than equations, there exists a non-zero solution (c₁, . . . , c_m+1) ∈ K₀(S)^m+1. The polynomialf :=Pm+1

i=1 c_i·t_i has the desired properties.

Linear Independence Techniques

This chapter deals with the theme of linear independence. First we present the Alternant Criterion for linear independence of polynomials. Using tech-niques from the existing literature, we give constructions of rank-preserving homomorphisms for linear forms, sparse polynomials, and products of linear forms. On the way, we encounter hitting set constructions for sparse poly-nomials and ΣΠΣ-circuits with constant top fan-in. Using isolating weight vectors, we generalize the hitting sets for sparse polynomials to polynomi-als whose Newton polytope can be decomposed into sparse polytopes. All constructions will be independent of the field of constants. Finally, we out-line that out-linear independence testing and the computation of out-linear rela-tions is (more or less) equivalent to PIT. In this context, we extend the polynomial-time PIT algorithm [RS05] for set-multilinear ΣΠΣ-circuits (with unbounded top fan-in) to an algorithm for computing the linear relations of set-multilinear ΠΣ-circuits.

Chapter outline

This chapter is organized as follows. Section 3.1 contains a criterion for lin-ear independence of polynomials. In Section 3.2 we define rank-preserving homomorphisms and give explicit constructions of rank-preserving homo-morphisms and hitting sets for several circuit classes. We summarize those results in Section 3.2.5. Section 3.3 deals with the linear independence testing problem. Finally, in Section 3.4, we investigate the complexity of computing linear relations.

41

3.1 Linear Independence

In this section we introduce a bit of notation connected with linear indepen-dence and present a criterion for linear indepenindepen-dence of polynomials.

LetK be a field, letAbe aK-vector space, and leta₁, . . . , a_m ∈A. Then LinRel_K(a₁, . . . , a_m) :=

λ∈K^m|λ₁a₁+· · ·+λ_ma_m = 0 (3.1.1) is a K-subspace of K^m and is called the subspace of linear relations of a₁, . . . , a_m over K. It is the kernel of the K-linear epimorphism

K^m → ha₁, . . . , a_mi_K, λ 7→λ₁a₁+· · ·+λ_ma_m. For a subset S⊆A, we define therank of S over K as

rk_K(S) := dim_K hSi_K

∈N∪ {∞}. (3.1.2) We are primarily interested in the case whereAis a polynomial ring overK.

3.1.1 The Alternant Criterion

Let K be a field and let K[x] = K[x₁, . . . , x_n] be a polynomial ring over K. The following theorem contains a criterion for linear independence of polynomials in K[x] if the field K is sufficiently large.

Theorem 3.1.1 (Alternant Criterion). Let K be an infinite field and let f1, . . . , fm ∈ K[x] be polynomials. Then f1, . . . , fm are K-linearly indepen-dent if and only if there exist points a₁, . . . ,a_m ∈Kⁿ such that

det fi(aj)

1≤i,j≤m 6= 0.

Proof. By Theorem 2.5.4, this follows from Lemma 3.1.2 below.

The Alternant Criterion is based on the following assertion which ap-peared in the proof of [Kay10, Lemma 8].

Lemma 3.1.2. Let f₁, . . . , f_m ∈K[x] be polynomials. Define the matrix

A:=







f₁(t_1,1, . . . , t_1,n) · · · f_m(t_1,1, . . . , t_1,n)

... ...

f₁(t_m,1, . . . , t_m,n) · · · f_m(t_m,1, . . . , t_m,n)





∈K[t]^m×m,

where t ={t_i,j| i∈[m] and j ∈[n]} are new variables. Then f₁, . . . , f_m are K-linearly independent if and only if det(A)6= 0.

Proof. By a linear algebra argument, f₁, . . . , f_m are K-linearly independent if and only if they are K-linearly independent. Therefore, we may assume that K is infinite.

First let f₁, . . . , f_m be K-linearly dependent. Then the columns ofA are K(t)-linearly dependent, hence det(A) = 0.

Conversely, assume thatf₁, . . . , f_m are K-linearly independent. We show det(A) 6= 0 by induction on m. The case m = 1 is obvious, so let m ≥ 2.

Expanding det(A) by the last row, we get det(A) =

m

X

j=1

(−1)^j+m·f_j(t_m,1, . . . , t_m,n)·det(A_m,j), (3.1.3)

whereA_m,j ∈K[t\{t_m,1, . . . , t_m,n}](m−1)×(m−1) is obtained fromAby deleting them-th row andj-th column. By induction hypothesis, we have det(A_m,1)6=

0. Since K is infinite, Theorem 2.5.4 implies that there exist c_i,k ∈ K for i∈[m−1] andj ∈[n] such that (det(A_m,1))(c)6= 0, where c= (c_i,k). Since f₁(t_m,1, . . . , t_m,n), . . . , f_m(t_m,1, . . . , t_m,n) are K-linearly independent, (3.1.3) implies (det(A))(c)6= 0, hence det(A)6= 0.

Im Dokument Independence in Algebraic Complexity Theory (Seite 44-53)