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Biman Roy

Publications and source records attributed to Biman Roy.

5 recordsLinked to original sources
arXiv2026

$\mathbb{A}^1$-fibration in algebraic geometry and $\mathbb{A}^1$-homotopy type

In this article we show that an $\mathbb{A}^1$ bundle map or a vector bundle map $p: X \to Y$ induces trivial local fibration $\underline{Sing}(X) \to \underline{Sing}(Y)$. Using this, we first show that for Korus Russel threefolds of first kind $X$ the space $\underline{Sing}(X)$ is $\mathbb{A}^1$ local. Then we show that for any smooth affine complex surface $X$, the $\mathbb{A}^1$- connected component sheaf is homotopy invariant.

math.AG↗
arXiv2026

Vector Bundles on Rational Topologically Contractible Affine Threefolds

The generalized Serre question asks whether every algebraic vector bundle on a topologically contractible smooth affine complex variety is trivial. We give an affirmative answer for rational threefolds. More generally, for a topologically contractible smooth affine complex threefold $X$, we prove that $\text{CH}^2(X)=0$ whenever $X$ admits a smooth projective compactification whose Chow group of $0$-cycles is supported on a curve. This uncovers the link between the generalized van de Ven question, Bloch's conjecture and the generalized Serre question for threefolds. We also prove that every Koras-Russell threefold is rational and therefore has only trivial algebraic vector bundles, hence answer a question of Koras and Russell.

math.AG↗
arXiv2024

$\mathbb{A}^1$-homotopy type of $\mathbb{A}^2 \setminus \left\{(0,0) \right\}$

In this article we prove that any $\mathbb{A}^1$-connected smooth $k$-variety is $\mathbb{A}^1$-uniruled for any algebraically closed field $k$. We establish that if a non empty open subscheme $X$ of a smooth affine $k$-scheme is $\mathbb{A}^1$-weakly equivalent to $\mathbb{A}^2_{k} \setminus \left\{(0,0) \right\}$, then $X \cong \mathbb{A}^2_{k} \setminus \left\{(0,0) \right\}$ as $k$-varieties for any field $k$ of characteristic $0$.

math.AG↗
arXiv2017

Time Complexity of Constraint Satisfaction via Universal Algebra

The exponential-time hypothesis (ETH) states that 3-SAT is not solvable in subexponential time, i.e. not solvable in O(c^n) time for arbitrary c > 1, where n denotes the number of variables. Problems like k-SAT can be viewed as special cases of the constraint satisfaction problem (CSP), which is the problem of determining whether a set of constraints is satisfiable. In this paper we study thef worst-case time complexity of NP-complete CSPs. Our main interest is in the CSP problem parameterized by a constraint language Gamma (CSP(Gamma)), and how the choice of Gamma affects the time complexity. It is believed that CSP(Gamma) is either tractable or NP-complete, and the algebraic CSP dichotomy conjecture gives a sharp delineation of these two classes based on algebraic properties of constraint languages. Under this conjecture and the ETH, we first rule out the existence of subexponential algorithms for finite-domain NP-complete CSP(Gamma) problems. This result also extends to certain infinite-domain CSPs and structurally restricted CSP(Gamma) problems. We then begin a study of the complexity of NP-complete CSPs where one is allowed to arbitrarily restrict the values of individual variables, which is a very well-studied subclass of CSPs. For such CSPs with finite domain D, we identify a relation SD such that (1) CSP({SD}) is NP-complete and (2) if CSP(Gamma) over D is NP-complete and solvable in O(c^n) time, then CSP({SD}) is solvable in O(c^n) time, too. Hence, the time complexity of CSP({SD}) is a lower bound for all CSPs of this particular kind. We also prove that the complexity of CSP({SD}) is decreasing when |D| increases, unless the ETH is false. This implies, for instance, that for every c>1 there exists a finite-domain Gamma such that CSP(Gamma) is NP-complete and solvable in O(c^n) time.

cs.CC↗