Multivariate Derivatives

Theorem 3.1 (Baur–Strassen). Fix a fieldFof characteristic 0, 0,1∈C⊆F, and let P denote a straight-line program in n variables overS= F,C,(+,−,×,÷)

computing f ∈F(X1, . . . ,X_n).

Then there exists a straight-line program P^′in n variables overSof length|P^′| ≤5· |P| simulta-neously computing all f,∂1f, . . . ,∂nf .

See [1,§7.2].

Lemma 3.2 (Taylor and Leibniz). For f ∈F(X1, . . . ,X_N)define

f⁽⁰⁾:= f(~0)∈F, f^(d) :=

∑

N n1,...nd=1

∂n1···∂n_df

(~0)·X_n1···X_nd/d! ∈ F[X1, . . .,X_N]=d

a) For f ∈F[X1, . . . ,X_N]<Dit holds f =∑^D_d=0⁻¹f^(d).

b) (f·g)⁽⁰⁾= (f·g)(~0) = f⁽⁰⁾·g⁽⁰⁾∈F, (f·g)⁽¹⁾= f⁽¹⁾·g⁽⁰⁾+ f⁽⁰⁾·g⁽¹⁾,

(f·g)⁽²⁾= f⁽²⁾·g⁽⁰⁾+f⁽¹⁾·g⁽¹⁾+f⁽⁰⁾·g⁽²⁾, and(f·g)^(D)=∑^D_d=0f^(d)·g^(D−d). c) In case g(~0)6=0, u := f/g has u⁽⁰⁾= f⁽⁰⁾/g⁽⁰⁾, u⁽¹⁾= f⁽¹⁾−u⁽⁰⁾·g⁽¹⁾

/g⁽⁰⁾, u⁽²⁾= f⁽²⁾−u⁽¹⁾·g⁽¹⁾−u⁽⁰⁾·g⁽²⁾

/g⁽⁰⁾, and u^(D)= f^(D)−∑^D_d=0⁻¹u^(d)·g^(D−d) /g⁽⁰⁾. Theorem 3.3 (Strassen’73). LetAdenote anF–algebra. SupposeF⊆F[X1, . . . ,X_N]<Dcan be computed (on a Zariski–dense subset ofA^N) by a straight-line program P over (A,C,+,×,÷) can also be computed by a straight-line program Q over(A,C,+,×)of length|Q| ≤O(D²)· |P|. See [1,§7.1].

3.2 Univariate Polynomial Arithmetic AbbreviateS^′:= C,C,+,×,÷

.

Theorem 3.4 (Polynomial Multiplication).

a) The product of two polynomials ¯p,q¯∈C[X], given by their lists of coefficients (dense repre-sentation), can be computed by a straight-line program overS^′of lengthO(N·log N), where N :=deg(p) +¯ deg(q).¯

b) The (coefficients of the) product of k given polynomials ¯p1, . . . ,p¯_k∈C[X]_<d,

can be computed by a straight-line program overS^′of lengthO(N·log²N), where N :=d·k.

See [1,§2.3].

Lemma 3.5. a) ¯p=∑n≥0p_nXⁿ∈F[[X]]has a multiplicative inverse 1/p¯∈F[[X]]iff p₀6=0;

in which case ¯q=∑n≥0q_nXⁿ:=1/p is given by q¯ ₀=1/p0and inductively q_n=−∑ⁿ_m=1p_m· qn−m/p0.

b) Suppose ˜q∈F[[X]]satisfies ¯p·q˜≡1 (mod Xⁿ). Then ˜˜q :=q˜·(2−p¯·q)˜ has ¯p·˜˜q≡1 (mod X²ⁿ).

c) Fix polynomials ¯a=∑ⁿ_i=0a_iXⁱ, ¯b=∑^m_j=0b_jX^j, ¯q=∑^m_k=0⁻ⁿq_kX^k, and ¯r=∑^m_ℓ=0⁻¹r_ℓX^ℓ with ¯a=¯b·q¯+¯r, where n :=deg(a)¯ ≥deg(¯b) =: m>deg(¯r). Then

∑

ⁿi=0a_iXⁿ⁻ⁱ

/

∑

^mj=0b_jX^m−j

≡

∑

^mk=0⁻ⁿq_kX^n−m−k (mod X^n−m+1)

d) For x₁, . . .,x_N ∈Fand ¯a∈F[X], ¯r :=a rem¯ (X−x₁)···(X−x_N)satisfies ¯a(xn) =¯r(xn).

e) It holds ¯a rem ¯p= (a rem ¯¯ p·q)¯ rem ¯p.

Theorem 3.6 (Polynomial Division and Multipoint Evaluation).

a) There exists a straight-line program over S^′ of length O(N·log N) computing, given (the coefficients of) p∈C[X]<N with p(0)6=0, (the coefficients of) 1/p mod X^N.

b) Given (the coefficients of) a,b∈C[X]of N :=deg(a)≥deg(b) =: M≥1, (the coefficients of) a div b and a rem b can be computed by a straight-line program overS^′of lengthO(N·log N).

c) A straight-line program overS^′ of lengthO(N·log²N)can compute, given (the coefficients of) p∈C[X]<N and x₁, . . .,x_N∈C, the values p(x1), . . .,p(xN).

d) A straight-line program overS^′ of length O Nd·log²(Nd)

can compute, given (the coef-ficients of) p₁, . . . ,p_N,q₁, . . .,q_N ∈C[X]_<d and z₁, . . . ,z_Nd ∈ C with q_j(zi)6=0, the values

∑^N_j=1^p_q^j_j^(z_(ziⁱ⁾₎, 1≤i≤Nd.

e) A straight-line program overS^′of lengthO(N·log N+log M)can compute, given p∈C[X]<N, p^M mod X^N.

See [6,§9+§10.10.1], [1,§2.4], and [10, THEOREM 2].

4 Complexity of Matrix Multiplication

4.1 Strassen’s Algorithm

4.2 Complexity and Tensor Rank of Bilinear Maps 4.3 Properties of the Tensor Rank

4.4 Exponent of Matrix Multiplication, LUP-Decomposition, and Inversion 4.5 Multipoint Evaluation of Bivariate Polynomials

5 Branching Complexity

Question 5.1 (Sorting) Given x₁, . . . ,x_nin a fixed linearly ordered set, how many comparisons are asymptotically sufficient and necessary to produce a permutationπ:[n]→[n] with x_π(1) ≤ x_π(2)≤ ··· ≤x_π(n) ?

• upper bound n·(n+1)/2: Bubble Sort

• upper boundO(n·log n): Merge Sort

• lower boundΩ(log₂n!) =O(n·log n):

Definition 5.2 (Decision Tree). Let S= S,R

denote a structure withRa family of relations R :⊆S^ak of arities a_R ∈N and Σsome arbitrary set. A Decision Tree T (over S andΣ and in variables X₁, . . . ,X_n) is an ordered full binary tree with each internal node u labelled by one of the above relations R_uand by an a_u:=a_Ru–tuple(Xu1, . . . ,X_uau)of the variables; while leaves v are labelled with elementsσv∈Σ.

When assigned values x1, . . . ,xn∈S to X1, . . . ,Xn, T starts at its root and for each internal node u iteratively proceeds to its left or right child depending on R_u(xu1, . . .,x_uau). Upon ending up in a leaf v it outputs T(x1, . . . ,x_n):=σv.

5.1 Randomized Polynomial Identity Testing

Definition 5.3. Polynomial Identity Testingis the following decision problem:

Given an expression p composed from variables X1, . . . ,Xnand integer constants using addition +and multiplication×; does this p represent the zero function onQ//R/C?

Any such expression p represents a multivariate integer polynomial; but expanding it into mono-mials can blow up its size:

For instance the determinant of a given n×n–matrix A= (ai j) is an n²–variate polynomial of total degree n in A’s entries. Expanded into monomials it consists of n! terms (Leibniz Formula) yet can be evaluated (on a Zariski–dense subset ofF^n×n) inO(n³) steps by means of Gaussian Elimination.

Lemma 5.4 (Schwartz,Zippel). LetFdenote a field, S⊆Ffinite, and p∈F[X1, . . . ,X_n]a non-zero polynomial of total degree d ∈N. Then, for r₁, . . .,r_n∈S chosen independently and uni-formly from S at random,

P[p(r1, . . . ,r_n) =0] ≤ d/|S| .

5.2 Recap on Semi-Algebraic Geometry Definition 5.5. Fix a ringF⊆Rand d∈N.

a) A set A of real solutions to a system of polynomial equalities (overF) is algebraic (overF):

~x∈R^d : p₁(~x) =. . .= p_k(~x) =0}, p₁, . . .,p_k∈F[X1, . . .,X_d] b) A constructible set is a finite Boolean combination of algebraic sets.

c) A set of solutions to a finite system of polynomial in-/equalities

~x∈R^d : p₁(~x) =. . .=p_k(~x) =0 ∧ q₁(~x)>0∧. . .∧q_ℓ(~x)>0 with p₁, . . . ,p_k,q₁, . . .,q_ℓ∈F[X1, . . . ,X_d]is called basic semi-algebraic (overF).

d) A subset ofR^dsemi-algebraic is a finite union of basic semi-algebraic ones.

e) It is countably semi-algebraic overFif the union involves countably many members, all being basic semi-algebraic overF.

Example 5.6 a) A circle is algebraic overZ. A disc is basic semi-algebraic overZ.

Every integer polytope is basic semi-algebraic overZ.

b) Every constructible subset ofRis finite or co-finite;

every semi-algebraic subset ofRis a finite union of intervals.

c) Every semi-algebraic set is the projection of a constructible set.

Fact 5.7 (Tarski–Seidenberg) The projection of a semi-algebraic set is again semi-algebraic!

5.3 Recap on Projective Geometry Definition 5.8. Fix a fieldF⊇Qand d∈N.

a) Projective spaceP^d(F)is the set{[~v]:~06=~v∈F^d+1}of lines through the origin, where[~v]:={λ~v :λ∈F}denotes a projective point.

b) The Grassmannian Gr_k(F^d)is the set of k–dimensional linear subspaces ofF^d; Gr(F^d):=^S_kGr_k(F^d). (So Gr₁(F^d+1) =P^d(F). . . )

c) For(~a1, . . .,~a_d)^†=B= (~b1, . . .,~b_k)∈F^d×k a matrix of full rank, the family of its maximal minors

Det span(B)

:= det(~a_i1, . . . ,~a_ik)

1≤i₁<i₂<...<i_k≤d

is called the Pl¨ucker Coordinates of span(B)∈Gr_k(F^d).

Lemma 5.9. Det : Gr_k(F^d)→P(^dk)−1

(F)is well-defined and injective (but not surjective).

See [17, PROPOSITION 14.2].

5.4 Ben-Or’s Lower Bound and Applications

5.5 Range Spaces and their Vapnik-Chervonenkis Dimension 5.6 Fast Point Location in Arrangements of Hyperplanes 5.7 Polynomial-depth Algorithms forNP–complete Problems

6 NP –Completeness over the Reals

A BCSS machine M (over R) can in each step add, subtract, multiply, divide, and branch on the result of comparing two reals. Its memory consists of an infinite sequence of cells, each capable of holding a real number and accessed through an index register (similar to a one-head Turing machine). A program for M may store a finite number of real constants. The notions of decidability and semi-decidability translate straightforwardly from discrete L⊆ {0,1}^∗ and L⊆N^∗to real languagesL⊆R^∗. Computing a function f :⊆R^∗→R^∗means that the machine, given~x ∈dom(f), outputs f(~x) within finitely many steps and terminates while diverging on inputs~x6∈dom(f).

Example 6.1 a) rank :R^n×m→Nis uniformly BCSS–computable in timeO(n³+m³) b) The graph of the square root function is BCSS–decidable.

c) Qis BCSS semi-decidable; and so is the setAof algebraic reals.

d) The algebraic degree function deg :A→Nis BCSS–computable.

e) A languageL⊆R^∗is BCSS semi-decidable iff L=range(f)for some total computable f :R^∗→R^∗. f) The real Halting problemHis not BCSS–decidable, where

H :=

hM,~xi: BCSS machineMterminates on input~x g) Every discrete language L⊆ {0,1}^∗is BCSS–decidable!

h) The following discrete problems (i) FEAS⁰_Rand (ii) QUART⁰_R can be verified in polynomial time by a BCSS machine without constants:

i) Given (the degrees and coefficients in binary of) a system of multivariate

polynomial in-/equalities with integer coefficients, does it admit a real solution?

ii) Given a multivariate polynomial of total degree at most 4, does it admit a real root?

Definition 6.2. LetNP⁰

Rdenote the family of discrete decision problems of the form ~x∈ {0,1}ⁿ: n∈N,∃~y∈R^p(n):h~x,~yi ∈V

where p∈N[N] and V⊆R^∗ can be decided in polynomial time by a BCSS machine without constants.

Theorem 6.3. FEAS⁰_Rand QUART⁰_Rare complete forNP⁰

R

(with respect to many-one reduction by a polynomial-time Turing machine).

Fact 6.4 (Grigoriev’88,Canny’88,Heintz&Roy&Solern´o’90,Renegar’92) NP⊆NP⁰

R⊆PSPACE.

6.1 Equations over the Cross Product

The cross product inR³is well-known due to its many applications in physics such as torque or electromagnetism. Mathematically it constitutes the mapping

Fact 6.5 a) For linearly independent~v,~w, their cross product~v×~w=:~u is uniquely determined by the following:~u⊥~v, ~u⊥~w (where “⊥” denotes orthogonality), the triplet~v,~w,~u is right-handed, and lengths satisfyk~uk=k~vk · k~wk ·cos∠(~v,~w).

In particular, anti-/parallel~v,~w are mapped to~0.

b) Cross products commute with simultaneous orientation preserving orthogonal transforma-tions: For O∈R^3×3 with O·O^†=id and det(O) =1 it holds(O·~v)×(O·~w) =O·(~v×~w), Following JOHN VON NEUMANN (who in turn credits KARL VONSTAUDT), express arithmetic overFas geometric operations onF³by identifying r∈Fwith the line _x

rx

b) F(~v1−s~v3) = V2×

V23×F(~v1−s~v2) c) F(~v₃−r~v₂) = V₁×

V₁₃×F(~v₁−r~v₂) d) F ~v1−(r−s)~v2

= V3×

[V23×F(~v1−r~v2)]×[V2×F(~v1−s~v3)]

×V3 e) V₁₃ = V₂×(V12×V₂₃).

f) For W ∈P²(F), the expression ı(W):= (W×V3)×

(W×V3)×V3

×V2

is defined pre-cisely when W =F(~v₁−r~v₂+s~v₃) for some s ∈F and a unique r∈ F; and in this case ı(W) =F(~v₁−r~v₂). Moreover, if W =F(~v₁−r~v₂)then ı(W) =W .

Theorem 6.9. a) XNONTRIV⁰_R3,XNONTRIV⁰_P2(R),XUVEC⁰_R3, andXNONEQUIV⁰_P2(R)are polytime equivalent toPolynomial Identity Testing(Definition 5.3).

b) XSAT⁰_R3 andXSAT⁰_P2(R) areNP⁰

R–complete.

c) There is a term t(V1, . . . ,Vn)s.t.~06=t(V1, . . .,Vn) =V1is satisfiable overR³but not overQ³.

6.2 Satisfiability in Quantum Logic

Definition 6.10. a) For a vector space V , the Grassmannian Gr_k(V)is the set of k–dimensional linear subspaces of V ;

Gr(V):=^S_kGr_k(V), 1 :=V is called (strong) truth, every X 6=0 :={~0}isweakly true.

b) For a finite-dimensional inner product space V , equip Gr(V)with the operations X∧Y :=X∩Y, X∨Y :=X+Y, and¬X :=X^⊥={~v∈V :∀~a∈X :~v⊥~a} . c) Alattice termis an expression over variables and∨,∧;

an (ortho)termmay in addition involve¬.

d) For a term t with variables X1, . . . ,Xn and for an assignment x1, . . . ,xn∈Gr(V), the value t_V(x1, . . . ,x_n)is defined inductively according to b).

We may omit the subscript V if it is clear from the context.

e) C(X,Y):= (X∧Y)∨(X∧ ¬Y)∨(¬X∧Y)∨(¬X∧ ¬Y)is calledcommutator(of X and Y ).

f) SAT_V :={ht(X1, . . . ,X_n)i:∃x₁, . . . ,x_n∈Gr(V): t_V(x1, . . . ,x_n) =1}, satV :={ht(X1, . . . ,Xn)i:∃x1, . . .,xn∈Gr(V): tV(x1, . . . ,xn)6=0}. g) For a term t(X1, . . .,X_n)and a fieldF⊆Clet

maxdim_F(t,d) := max

dim t_Fd(x1, . . .,x_n)

: x₁, . . .,x_n∈Gr(F^d) .

h) A d–diamondin V is a(d+1)–tuple D0,D₁, . . .,D_d∈Gr(V)such that V =D₁^k. . .^kD_d= D₀⊕D_jfor all 1≤ j≤d, where^kand⊕denote orthogonal and direct sum, respectively.

SoSAT_F1 =sat_F1 coincides with the classical, Boolean satisfiability problem;

hti ∈SAT_F^d ⇔maxdim_F(t,d) =d,hti ∈sat_F^d ⇔maxdim_F(t,d)>0.

Lemma 6.11. a) Gr(V)satisfiesde Morgan’s Rules¬(X∨Y) = (¬X)∧(¬Y) and ¬(X∧ Y) = (¬X)∨(¬Y); but Gr(R²)violates the distributive law(X∨Y)∧Z= (X∧Z)∨(Y∧Z).

b) Gr(V)satisfies themodular laws

g) Conversely every choice of E_i,E_{i j}∈Gr(F^d)satisfying the conditions expressed in f) arise from a basis e_i.

Theorem 6.13. a) For any fieldF⊆C, bothSAT_F² andsat_F² areNP–complete.

b) For every d≥3,SAT_R^d andsat_R^d areNP_R–complete c) and so areSAT_Cd andsat_Cd!

d) There exists a term t that is (weakly/strongly) satisfiable over Gr(R³)but not over Gr(Q³) and a term s (weakly/strongly) satisfiable over Gr(C³)but not over Gr(R³).

Lemma 6.14. a) For terms s(X1, . . . ,Xn)and t(Y1, . . .,Ym)it holds

e) For d :=dim(V), every weakly satisfying assignment D0,D1, . . . ,D_d∈Gr(V)of Equation (2) constitutes a d-diamond.

Moreover, in this case, g_d,j(D0,D₁, . . . ,D_d) =D_j and dim g_d(D0,D₁, . . . ,D_d)

=1.

f) If t is weakly satisfiable over Gr(V), there exists some W ∈Gr_|t|(V) such that t is weakly satisfiable over Gr(W), where|t|denotes the syntactic length of t.

Definition 6.15. Call t(X1, . . .,X_n) weakly/strongly satisfiable over Gr(F^∗) if there exists some d∈Nand a weakly/strongly satisfying assignment x1, . . .,xn∈Gr(F^d).

6.3 Realizability of Oriented Matroids 6.4 Stretchability of Pseudolines

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Im Dokument Script Skeleton: Algebraic Complexity Theory (Seite 6-0)