8 research outputs found
New Lower Bounds Against Homogeneous Non-Commutative Circuits
We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree d which requires homogeneous non-commutative circuit of size ?(d/log d). For an n-variate polynomial with n > 1, the result can be improved to ?(nd), if d ? n, or ?(nd (log n)/(log d)), if d ? n. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial
Separating ABPs and Some Structured Formulas in the Non-Commutative Setting
The motivating question for this work is a long standing open problem, posed
by Nisan (1991), regarding the relative powers of algebraic branching programs
(ABPs) and formulas in the non-commutative setting. Even though the general
question continues to remain open, we make some progress towards its
resolution. To that effect, we generalise the notion of ordered polynomials in
the non-commutative setting (defined by \Hrubes, Wigderson and Yehudayoff
(2011)) to define abecedarian polynomials and models that naturally compute
them.
Our main contribution is a possible new approach towards separating formulas
and ABPs in the non-commutative setting, via lower bounds against abecedarian
formulas. In particular, we show the following.
There is an explicit n-variate degree d abecedarian polynomial
such that 1. can be computed by an abecedarian ABP of size O(nd);
2. any abecedarian formula computing must have size that is
super-polynomial in n.
We also show that a super-polynomial lower bound against abecedarian formulas
for would separate the powers of formulas and ABPs in the
non-commutative setting
Constructing Faithful Homomorphisms over Fields of Finite Characteristic
We study the question of algebraic rank or transcendence degree preserving homomorphisms over finite fields. This concept was first introduced by Beecken et al. [Malte Beecken et al., 2013] and exploited by them and Agrawal et al. [Manindra Agrawal et al., 2016] to design algebraic independence based identity tests using the Jacobian criterion over characteristic zero fields. An analogue of such constructions over finite characteristic fields were unknown due to the failure of the Jacobian criterion over finite characteristic fields.
Building on a recent criterion of Pandey, Saxena and Sinhababu [Anurag Pandey et al., 2018], we construct explicit faithful maps for some natural classes of polynomials in fields of positive characteristic, when a certain parameter called the inseparable degree of the underlying polynomials is bounded (this parameter is always 1 in fields of characteristic zero). This presents the first generalisation of some of the results of Beecken, Mittmann and Saxena [Malte Beecken et al., 2013] and Agrawal, Saha, Saptharishi, Saxena [Manindra Agrawal et al., 2016] in the positive characteristic setting
Constructing Faithful Homomorphisms over Fields of Finite Characteristic
We study the question of algebraic rank or transcendence degree preserving
homomorphisms over finite fields. This concept was first introduced by Beecken,
Mittmann and Saxena (Information and Computing, 2013), and exploited by them,
and Agrawal, Saha, Saptharishi and Saxena (Journal of Computing, 2016) to
design algebraic independence based identity tests using the Jacobian criterion
over characteristic zero fields. An analogue of such constructions over finite
characteristic fields was unknown due to the failure of the Jacobian criterion
over finite characteristic fields.
Building on a recent criterion of Pandey, Sinhababu and Saxena (MFCS, 2016),
we construct explicit faithful maps for some natural classes of polynomials in
the positive characteristic field setting, when a certain parameter called the
inseparable degree of the underlying polynomials is bounded (this parameter is
always 1 in fields of characteristic zero). This presents the first
generalisation of some of the results of Beecken et al. and Agrawal et al. in
the positive characteristic setting
On Annihilators of Explicit Polynomial Maps
We study the algebraic complexity of annihilators of polynomials maps. In
particular, when a polynomial map is `encoded by' a small algebraic circuit, we
show that the coefficients of an annihilator of the map can be computed in
PSPACE. Even when the underlying field is that of reals or complex numbers, an
analogous statement is true. We achieve this by using the class VPSPACE that
coincides with computability of coefficients in PSPACE, over integers.
As a consequence, we derive the following two conditional results. First, we
show that a VP-explicit hitting set generator for all of VP would separate
either VP from VNP, or non-uniform P from PSPACE. Second, in relation to
algebraic natural proofs, we show that proving an algebraic natural proofs
barrier would imply either VP VNP or DSPACE()
P
Monotone Classes Beyond VNP
We study the natural monotone analogues of various equivalent definitions of
VPSPACE: a well studied class (Poizat '08, Koiran-Perifel '09, Malod '11,
Mahajan-Rao '13) that is believed to be larger than VNP. We show an exponential
separation between the monotone version of Poizat's definition, and monotone
VNP. We also show that unlike their non-monotone counterparts, these monotone
analogues are not equivalent, with exponential separations in some cases.
The primary motivation behind our work is to understand the monotone
complexity of transparent polynomials, a concept that was recently introduced
by Hrube\v{s} and Yehudayoff (2021). In that context, we are able to show that
transparent polynomials of large sparsity are hard for the monotone analogues
of all known definitions of VPSPACE, except for the one due to Poizat.Comment: 26 pages; the draft has been shortened and simplified to now focus
solely on monotone classes beyond VN