TY - JOUR

T1 - Bounding the equivariant Betti numbers of symmetric semi-algebraic sets

AU - Basu, Saugata

AU - Riener, Cordian

PY - 2017/1/10

Y1 - 2017/1/10

N2 - Let R be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semi-algebraic subsets of Rk in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to Oleĭnik and Petrovskiĭ, Thom and Milnor. These bounds are all exponential in the number of variables k. Motivated by several applications in real algebraic geometry, as well as in theoretical computer science, where such bounds have found applications, we consider in this paper the problem of bounding the equivariant Betti numbers of symmetric algebraic and semi-algebraic subsets of Rk. We obtain several asymptotically tight upper bounds. In particular, we prove that if S⊂Rk is a semi-algebraic subset defined by a finite set of s symmetric polynomials of degree at most d, then the sum of the Sk-equivariant Betti numbers of S with coefficients in Q is bounded by (skd)O(d). Unlike the classical bounds on the ordinary Betti numbers of real algebraic varieties and semi-algebraic sets, the above bound is polynomial in k when the degrees of the defining polynomials are bounded by a constant. As an application we improve the best known bound on the ordinary Betti numbers of the projection of a compact algebraic set improving for any fixed degree the best previously known bound for this problem due to Gabrielov, Vorobjov and Zell.

AB - Let R be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semi-algebraic subsets of Rk in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to Oleĭnik and Petrovskiĭ, Thom and Milnor. These bounds are all exponential in the number of variables k. Motivated by several applications in real algebraic geometry, as well as in theoretical computer science, where such bounds have found applications, we consider in this paper the problem of bounding the equivariant Betti numbers of symmetric algebraic and semi-algebraic subsets of Rk. We obtain several asymptotically tight upper bounds. In particular, we prove that if S⊂Rk is a semi-algebraic subset defined by a finite set of s symmetric polynomials of degree at most d, then the sum of the Sk-equivariant Betti numbers of S with coefficients in Q is bounded by (skd)O(d). Unlike the classical bounds on the ordinary Betti numbers of real algebraic varieties and semi-algebraic sets, the above bound is polynomial in k when the degrees of the defining polynomials are bounded by a constant. As an application we improve the best known bound on the ordinary Betti numbers of the projection of a compact algebraic set improving for any fixed degree the best previously known bound for this problem due to Gabrielov, Vorobjov and Zell.

KW - Equivariant Betti numbers

KW - Semi-algebraic sets

KW - Symmetric polynomials

UR - http://www.scopus.com/inward/record.url?scp=84991628484&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2016.09.015

DO - 10.1016/j.aim.2016.09.015

M3 - Article

AN - SCOPUS:84991628484

SN - 0001-8708

VL - 305

SP - 803

EP - 855

JO - ADVANCES IN MATHEMATICS

JF - ADVANCES IN MATHEMATICS

ER -