Real Root Finding for Equivariant Semi-Algebraic Systems ISSAC 2018 - - PowerPoint PPT Presentation

Real Root Finding for Equivariant Semi-Algebraic Systems

ISSAC 2018 Cordian Riener1 Mohab Safey El Din2

1UiT - The Arctic University of Norway 2Sorbonne Universit´

e, CNRS, INRIA Laboratoire d’Informatique de Paris 6, LIP6, ´ Equipe PolSys

July 9, 2018

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 1 / 17

Introduction

The objects we are looking at and the questions one can ask

In the sequel we will use a real closed field R

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 2 / 17

Introduction

The objects we are looking at and the questions one can ask

In the sequel we will use a real closed field R a finite set P ⊂ R[X1, . . . , Xk] defining a real variety ZR(P) ⊂ Rk,

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 2 / 17

Introduction

The objects we are looking at and the questions one can ask

In the sequel we will use a real closed field R a finite set P ⊂ R[X1, . . . , Xk] defining a real variety ZR(P) ⊂ Rk,

• r more generally a semi-algebraic S ⊂ Rk, which can be described by P.

Algorithmic problems associated to these, like, e.g., real root finding, connectivity queries, quantifier elimination, are known to be intrinsically hard.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 2 / 17

Symmetry

Philosophy: structure in a set can reduce complexity

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 3 / 17

How to exploit Symmetry ?

Do general symmetric problems have symmetric solutions?

YES, this must be the case! NO, this is just so wrong!

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 4 / 17

How to exploit Symmetry ?

Do general symmetric problems have symmetric solutions?

YES, this must be the case! NO, this is just so wrong! Proposition

Let S ⊂ Rn be basic convex symmetric semi-algebraic set. Then S is not empty if and only if it contains a point for which all coordinates are equal.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 4 / 17

What to do in a non convex situation?

Previous results

Theorem

Let {f , f1, . . . , fs} ⊂ R[x1, . . . , xn] be Symn-invariant polynomials of degree at most d.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 5 / 17

What to do in a non convex situation?

Previous results

Theorem

Let {f , f1, . . . , fs} ⊂ R[x1, . . . , xn] be Symn-invariant polynomials of degree at most d.

1

The real algebraic set V(f ) is not empty if and only if it contains a point with at most ⌊ d

2 ⌋ distinct coordinates.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 5 / 17

What to do in a non convex situation?

Previous results

Theorem

Let {f , f1, . . . , fs} ⊂ R[x1, . . . , xn] be Symn-invariant polynomials of degree at most d.

1

The real algebraic set V(f ) is not empty if and only if it contains a point with at most ⌊ d

2 ⌋ distinct coordinates.

2

The semi-algebraic set in S ⊂ Rn defined by f1 ≥ 0, . . . , fs ≥ 0 is not empty if and only if it contains a point with at most d distinct coordinates. This result helps to drastically reduce complexity and in particular it allows for a polynomial (in n) algorithm to test, if a given system described by symmetric polynomials of fixed degree has a solution. But what can we say in more general situations?

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 5 / 17

A more general situation

Definition

Consider G = (g1, . . . , gn) a sequences of polynomials in R[x1, . . . , xn]. If for all σ ∈ Sn, we have G(σ(X)) = σ(G)(X), then G is called equivariant.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 6 / 17

A more general situation

Definition

Consider G = (g1, . . . , gn) a sequences of polynomials in R[x1, . . . , xn]. If for all σ ∈ Sn, we have G(σ(X)) = σ(G)(X), then G is called equivariant.

Example

Let s be a bivariate symmetric polynomial and d ∈ N. Consider G = (xd

1 + s(x2, x3), xd 2 + s(x1, x3), xd 3 + s(x1, x2)). Then G is equivariant by the

action of the symmetric group Sym3. Now, a set of the form {x ∈ Rn : gi(x) ≥ 0 ∀g ∈ G} is clearly Symn-invariant, but does not directly fit into the pervious setup.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 6 / 17

Main geometric result

We consider F = (f1, . . . , fk) and G = (g1, . . . , gn) sequences of polynomials in R[x1, . . . , xn]. Further, let d be the maximum of deg(fi) and deg(gj) for 1 ≤ i ≤ k and 1 ≤ j ≤ n. Denote by S(F, G) the semi-algebraic set defined by f1 = · · · = fk = 0, g1 ≥ 0, . . . , gn ≥ 0.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 7 / 17

Main geometric result

We consider F = (f1, . . . , fk) and G = (g1, . . . , gn) sequences of polynomials in R[x1, . . . , xn]. Further, let d be the maximum of deg(fi) and deg(gj) for 1 ≤ i ≤ k and 1 ≤ j ≤ n. Denote by S(F, G) the semi-algebraic set defined by f1 = · · · = fk = 0, g1 ≥ 0, . . . , gn ≥ 0.

Theorem

Assume that for 1 ≤ i ≤ k, fi is Symn invariant, that G is Symn-equivariant and that deg(gj) ≥ 2 for 1 ≤ j ≤ n. Then, the basic semi-algebraic set S(F, G) is empty if and only if S(F, G) contains a point contains a point with at most 2d − 1 distinct coordinates.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 7 / 17

Ideas of the proof

Deciding emptiness

The proof uses three main ingredients: The fact that every symmetric polynomial can be uniquely written in terms of Newton sums (pi := X i

1 + X i 2 + . . . + X i n),

a consequence of a representation result by Shchwartsman, an optimization problem.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 8 / 17

Ideas of the proof

Deciding emptiness

The proof uses three main ingredients: The fact that every symmetric polynomial can be uniquely written in terms of Newton sums (pi := X i

1 + X i 2 + . . . + X i n),

a consequence of a representation result by Shchwartsman, an optimization problem.

Proposition (Shchwartsman)

Let G be a sequence of equivariant polynomialslet deg gi ≤ d. Then gi =

d

• j=0

sj · xj

i ,

where sj ∈ R[x1, . . . , xn]Symn is symmetric and of degree ≤ d − j + 1.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 8 / 17

Ideas of the proof

Suppose that S := S(F, G) is not empty and let y ∈ S. We set p1(y) = γ1, . . . , pd(y) = γd and define Nγ := {x ∈ Rn : p1(x) = γ1, . . . , pd(x) = γd}.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 9 / 17

Ideas of the proof

Suppose that S := S(F, G) is not empty and let y ∈ S. We set p1(y) = γ1, . . . , pd(y) = γd and define Nγ := {x ∈ Rn : p1(x) = γ1, . . . , pd(x) = γd}.

Proposition

Every symmetric polynomial f of degree ≤ d is constant on Nγ.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 9 / 17

Ideas of the proof

Suppose that S := S(F, G) is not empty and let y ∈ S. We set p1(y) = γ1, . . . , pd(y) = γd and define Nγ := {x ∈ Rn : p1(x) = γ1, . . . , pd(x) = γd}.

Proposition

Every symmetric polynomial f of degree ≤ d is constant on Nγ.

Lemma

Let d ≤ n, γ ∈ Rd. Consider (g1, . . . , gn) a sequence of polynomials of degree at most d in R[x1, . . . , xn] which are Symn- equivariant and ξ = (ξ1, . . . , ξn) ∈ Nγ. Then, there exist {α1, . . . , αt} ⊂ R with t ≤ d − 1 such that gi(ξ) = 0 if and only if ξi ∈ {α1, . . . , αt}.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 9 / 17

Main step of the proof

Consider S′ := S ∩ Nγ. Then, S′ is closed and bounded. Thus there exists a maximizer ξ = (ξ1, . . . , ξn) for the function pd+1 on S′. If gi(ξ) > 0 for all i, then ∇pd+1(ξ) = d

k=1 λk∇pk(ξ) (Lagrange) and thus

(d + 1)ξd

i = d

• k=1

λk(k − 1)ξ(

i k − 1),

and thus |{ξ1, . . . , xn}| ≤ d.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 10 / 17

Main step of the proof

Consider S′ := S ∩ Nγ. Then, S′ is closed and bounded. Thus there exists a maximizer ξ = (ξ1, . . . , ξn) for the function pd+1 on S′. If gi(ξ) > 0 for all i, then ∇pd+1(ξ) = d

k=1 λk∇pk(ξ) (Lagrange) and thus

(d + 1)ξd

i = d

• k=1

λk(k − 1)ξ(

i k − 1),

and thus |{ξ1, . . . , xn}| ≤ d. If gi(ξ) = 0 for one i use the Lemma above to conclude that there are at most d − 1 choices for ξi.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 10 / 17

Two algorithmic consequences

Deciding emptiness

Theorem

Let F and G be as above and d be an integer bounding the degrees of the polynomials in F and G. Assume that the polynomials in F are Symn-invariant and that G is Symn-equivariant and that d ≤ n/2. There exists an algorithm which, on input (F, G) decides whether S(F, G) is empty using at most nO(d) arithmetic operations in Q.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 11 / 17

Two algorithmic consequences

One block quantifier elimination

Now let F = (f1, . . . , fk) and G = (g1, . . . , gn) in Q[x1, . . . , xn, y1, . . . , yt] such that Symn acts on (x1, . . . , xn).

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 12 / 17

Two algorithmic consequences

One block quantifier elimination

Now let F = (f1, . . . , fk) and G = (g1, . . . , gn) in Q[x1, . . . , xn, y1, . . . , yt] such that Symn acts on (x1, . . . , xn). We consider the problem of computing a semi-algebraic description of the projection on the (y1, . . . , yt)-space of the set S(F, G) ⊂ Rn × Rt defined by f1 = · · · = fk = 0, g1 ≥ 0, . . . , gn ≥ 0,

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 12 / 17

Two algorithmic consequences

One block quantifier elimination

Now let F = (f1, . . . , fk) and G = (g1, . . . , gn) in Q[x1, . . . , xn, y1, . . . , yt] such that Symn acts on (x1, . . . , xn). We consider the problem of computing a semi-algebraic description of the projection on the (y1, . . . , yt)-space of the set S(F, G) ⊂ Rn × Rt defined by f1 = · · · = fk = 0, g1 ≥ 0, . . . , gn ≥ 0,i.e., solve the one-block quantifier elimination problem: Φ : ∃ ∈ Rn f1 = · · · = fk = 0, g1 ≥ 0, . . . , gn ≥ 0, (1) hence computing a quantifier-free formula which is equivalent to the quantified formula Φ.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 12 / 17

Two algorithmic consequences

One block quantifier elimination

Theorem

Let F, G and Φ be as above, d be the maximum degree in the variables in (x1, . . . , xn) of the entries of F and G and assume that d ≤ n

• 2. Then, there exists

a quantifier-free formula Ψ(Y ) = ∪K

k=1Ψk(Y ) which is equivalent to Φ, and such

that K ≤ nO(d) and: Ψk(Y ) = ∨ℓk

i=1 ∧ℓi,k j=1 (∨ℓi,j,k u=1 sign(ϕi,j,u,k) = σi,j,h,k)

with σi,j,h,k ∈ {0, 1, −1}, ℓk ≤ (n + k)d+1nO(dt) ℓi,k ≤ (n + k)d+1nO(d), ℓi,j,k ≤ nO(d) and the degrees of the polynomials ϕi,j,u are bounded by nd. Moreover, there exists an algorithm which computes Ψ using at most (k + n)dtnO(dt) arithmetic

• perations in Q.
• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 13 / 17

Experimental results

We take random dense systems of k symmetric polynomials and equivariant families in Q[x1, . . . , xn] both of degree d. n d k RagLib-Sym RagLib 5 3 3 1762 1779 6 3 3 1583 376822 7 3 3 3135

• 8

3 3 4344

• 5

3 4 0.4 0.4 6 3 4 0.4 21 7 3 4 0.6 440 8 3 4 0.9 11686

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 14 / 17

Experimental results

A problem of the IMO

Show that for a < b, the semi-algebraic set defined by m2 > m2

1

(a + b)2 4ab , (b − x1)(x1 − a) ≥ 0, . . . , (b − xn)(xn − a) ≥ 0 is empty with m1 = 1 n

n

• i=1

xi, m2 = 1 n

n

• i=1

x2

i ,

a = 1, b = 2. n 3 4 5 6 7 8 9 RAG-S 0.12 0.14 0.3 0.5 0.6 0.74 1.2 RAG 0.2 0.3 0.5 1.6 9.8 131 1978 Maple 0.2 14.7

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 15 / 17

More general results?

Can we hope for a result for general semi-algebraic set?

Example

Let f := n

i=1(xi − i)2 and its Symn orbit which we denote by F. Let S be the

semi-algebraic set {x ∈ Rn : ∃g ∈ F with g(x) = 0}. By construction, S is a finite set which coincides with the orbit of ξ = (1, . . . , n). Therefore, all points in S have distinct coordinates, but S is described by quadratic polynomials.

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 16 / 17

Thank you for your attention and next year in Tromsø!

• C. Riener, M. Safey El Din

Equivariant Semi-Algebraic Systems July 9, 2018 17 / 17