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Algebraic Complexity Theory SS 2014, Exercise Sheet #9

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Martin Ziegler Issued on 2014-06-17 Solutions due: 2014-06-23

Algebraic Complexity Theory SS 2014, Exercise Sheet #9

EXERCISE 17:

Referring to Definition 6.7 in the script:

a) Prove thatXNONTRIV0P2(R) andXNONTRIV0R3 andXUVEC0R3 are equal as sets.

b) Construct polynomial-time reductions fromXNONEQUIV0P2(R)toXNONTRIV0R3and back.

(Hint:[~s]6= [~t]⇔~s×~t6=~0).

c) Construct a polynomial-time reduction

fromXNONTRIV0R3 toPolynomial Identity Testing.

d) Verify that~t:= (~t×~w)× (~t×~w)×~t

is a multiple of~t for any~w non-parallel to~t.

Can every multiple of~t be obtained in this way?

e) Prove that~s:=

~w×(~s×~w)

×~s

× ~s×(~s×~w)

is a multiple of~s;

and every multiple of~s can be obtained in this way for some appropriate~w.

f) Construct a polynomial-time reduction fromXSAT0P2(R) toXSAT0R3.

EXERCISE 18:

Fix a subfieldFofR, such asQand recall thatP2(F) ={[~v]:~06=~v∈F3}

denotes the projective space overF3, where[~v]:={λ~v :λ∈F}. To A,B,C∈P2(F)consider V12:=B V2:= (A×B)×A V23 :=C×A V1:=V2×V23 V3:= V23×(B×V2)

×B (1) a) The evaluation of these terms may be undefined for some assignments of A,B,C. Verify that,

on the other hand, A := [~v1], B := [~v2−~v1]and C := [~v2+~v3], do evaluate — namely to V1= [~v1], V2= [~v2], V3= [~v3], V12= [~v1−~v2], V23= [~v2−~v3]

— for any choice of pairwise orthogonal non-zero~v1,~v2,~v3∈F3.

b) Conversely when all quantities in Equation (1) are defined, then V1=A and there exists a right-handed (!) orthogonal basis~v1,~v2,~v3 ofF3 such that Vj = [~vj]and V12= [~v1−~v2]and V23 = [~v2−~v3].

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