NP-hardness of testing equivalence to sparse polynomials and to constant-support polynomials
dc.contributor.author | Baraskar, Omkar Bhalchandra | |
dc.date.accessioned | 2025-07-22T19:16:02Z | |
dc.date.available | 2025-07-22T19:16:02Z | |
dc.date.issued | 2025-07-22 | |
dc.date.submitted | 2025-07-17 | |
dc.description.abstract | Given a list of monomials of a n-variate polynomial f over a field F, and an integer s, decide whether there exists an invertible transform A and a b such that f(Ax + b) has less than s monomials. This problem is called the Equivalence testing to sparse polynomials (ETsparse). It was studied in [GrigorievK93] over Q, in this work, they give an exponential in n^4 time algorithm for the problem. The lack of progress in the complexity of the problem over last three decades raises a question, is ETsparse hard? In this thesis we give an affirmative answer to the question by showing that it is NP-hard over any field. Sparse orbit complexity of a polynomial f is the smallest integer s_0 such that there exists an invertible transform A such that f(Ax) has s_0 monomials. Since ETsparse is NP-hard hence computing the sparse orbit complexity is also NP-hard. We also show that approximating the sparse orbit complexity upto a factor of s_f^{1/3-\epsilon} for any \epsilon \in (0,1/3) is NP-hard, where s_f is the number of monomials in f. Interestingly, this approximation result has been shown without invoking the celebrated PCP theorem. [ChillaraGS23] study a variant of the problem which focus on shift equivalence. More precisely, given f over some ring R (the input has the same representation as in ETsparse) and an integer s, does there exists a b such that f(x + b) has less than s monomials. It is called the SETsparse problem, [ChillaraGS23] showed that SETsparse is NP-hard when R is an integral domain which is not a field; we extend their result to the case when R is a field. Finally, we also study the problem of testing equivalence to constant-support polynomials; more precisely, given a polynomial f as before and with support \sigma, does there exists an invertible transform A such that f(Ax) has support \sigma -1. We call this problem ETsupport. We show that ETsupport is NP-hard for \sigma >= 5 and over any field. | |
dc.identifier.uri | https://hdl.handle.net/10012/22043 | |
dc.language.iso | en | |
dc.pending | false | |
dc.publisher | University of Waterloo | en |
dc.subject | complexity theory | |
dc.subject | algebraic complexity | |
dc.subject | equivalence testing | |
dc.title | NP-hardness of testing equivalence to sparse polynomials and to constant-support polynomials | |
dc.type | Master Thesis | |
uws-etd.degree | Master of Mathematics | |
uws-etd.degree.department | David R. Cheriton School of Computer Science | |
uws-etd.degree.discipline | Computer Science | |
uws-etd.degree.grantor | University of Waterloo | en |
uws-etd.embargo.terms | 0 | |
uws.contributor.advisor | Oliveira, Rafael | |
uws.contributor.advisor | Schost, Eric | |
uws.contributor.affiliation1 | Faculty of Mathematics | |
uws.peerReviewStatus | Unreviewed | en |
uws.published.city | Waterloo | en |
uws.published.country | Canada | en |
uws.published.province | Ontario | en |
uws.scholarLevel | Graduate | en |
uws.typeOfResource | Text | en |