On Projections of Semi-Algebraic Sets Defined by Few Quadratic Inequalities

نویسندگان

  • Saugata Basu
  • Thierry Zell
چکیده

Let S ⊂ Rk+m be a compact semi-algebraic set defined by P1 ≥ 0, . . . , P` ≥ 0, where Pi ∈ R[X1, . . . , Xk, Y1, . . . , Ym], and deg(Pi) ≤ 2, 1 ≤ i ≤ `. Let π denote the standard projection from Rk+m onto Rm. We prove that for any q > 0, the sum of the first q Betti numbers of π(S) is bounded by (k +m). We also present an algorithm for computing the the first q Betti numbers of π(S), whose complexity is (k +m) O(q`) . For fixed q and `, both the bounds are polynomial in k +m.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Algorithmic and topological aspects of semi-algebraic sets defined by quadratic polynomial

In this thesis, we consider semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials. Semi-algebraic sets of $R^k$ are defined as the smallest family of sets in $R^k$ that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the boolean operations (complementation, finite unions and finite intersections). We ...

متن کامل

Computing the Betti Numbers of Semi-algebraic Sets Defined by Partly Quadratic Systems of Polynomials

Let R be a real closed field, Q ⊂ R[Y1, . . . , Y`, X1, . . . , Xk], with degY (Q) ≤ 2, degX(Q) ≤ d,Q ∈ Q,#(Q) = m, and P ⊂ R[X1, . . . , Xk] with degX(P ) ≤ d, P ∈ P,#(P) = s. Let S ⊂ R`+k be a semi-algebraic set defined by a Boolean formula without negations, with atoms P = 0, P ≥ 0, P ≤ 0, P ∈ P ∪ Q. We describe an algorithm for computing the the Betti numbers of S generalizing a similar alg...

متن کامل

Computing the Top Betti Numbers of Semi-algebraic Sets Defined by Quadratic Inequalities in Polynomial Time

For any l > 0, we present an algorithm which takes as input a semi-algebraic set, S, defined by P1 ≤ 0, . . . , Ps ≤ 0, where each Pi ∈ R[X1, . . . ,Xk ] has degree ≤ 2, and computes the top l Betti numbers of S, bk−1(S), . . . , bk−l(S), in polynomial time. The complexity of the algorithm, stated more precisely, is ∑l+2 i=0 ( s i ) k2 O(min(l,s)) . For fixed l, the complexity of the algorithm ...

متن کامل

A Sharper Estimate on the Betti Numbers of Sets Defined by Quadratic Inequalities

In this paper we consider the problem of bounding the Betti numbers, bi(S), of a semi-algebraic set S ⊂ R k defined by polynomial inequalities P1 ≥ 0, . . . , Ps ≥ 0, where Pi ∈ R[X1, . . . , Xk] , s < k, and deg(Pi) ≤ 2, for 1 ≤ i ≤ s. We prove that for 0 ≤ i ≤ k − 1, bi(S) ≤ 1 2 + (k − s) + 1 2 · min{s+1,k−i}

متن کامل

Bounding the Number of Stable Homotopy Types of a Parametrized Family of Semi-algebraic Sets Defined by Quadratic Inequalities

We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in R, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in `. More precisely, we prove the following. Let R be a real closed field and let P = {P1, . . . , Pm} ⊂ R[Y1, . . . , Y`,X1, . . . , Xk], with degY (Pi) ≤ 2,d...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • Discrete & Computational Geometry

دوره 39  شماره 

صفحات  -

تاریخ انتشار 2008