Related papers: Lech's conjecture in dimension three

The long standing Lech's conjecture in commutative algebra states that for a flat local extension $(R,\mathfrak{m})\to (S,\mathfrak{n})$ of Noetherian local rings, we have an inequality on the Hilbert--Samuel multiplicities: $e(R)\leq…

We explore the classical Lech's inequality relating the Hilbert--Samuel multiplicity and colength of an $\mathfrak{m}$-primary ideal in a Noetherian local ring $(R,\mathfrak{m})$. We prove optimal versions of Lech's inequality for…

We study conjectured generalizations of a formula of Lech which relates the multiplicity of a finite colength ideal in an equicharacteristic local ring to its colength, and prove one of these generalizations involving the multiplicity of…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d\geq 2$. We prove that if $e(\widehat{R}_{red})>1$, then the classical Lech's inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists…

We introduce the notion of strongly Lech-independent ideals as a generalization of Lech-independent ideals defined by Lech and Hanes, and use this notion to derive inequalities on multiplicities of ideals. In particular we prove that if…

Let $(R,\mathfrak{m})$ be a Noetherian local ring, and let $M$ be a finitely generated $R$-module of dimension $d$. We prove that the set $\left\{\frac{l(M/IM)}{e(I, M)} \right\}_{\sqrt{I}=\mathfrak{m}}$ is bounded below by…

Let $S\subset\Ps^r$ ($r\geq 5$) be a nondegenerate, irreducible, smooth, complex, projective surface of degree $d$. Let $\delta_S$ be the number of double points of a general projection of $S$ to $\Ps^4$. In the present paper we prove that…

The Blaschke's conjecture asserts that if $\diam(M)=\text{Inj}(M)=\frac\pi2$ (up to a rescaling) for a complete Riemannian manifold $M$, then $M$ is isometric to $\Bbb S^n(\frac12)$, ${\Bbb R\Bbb P}^{n}$, ${\Bbb C\Bbb P}^{n}$, ${\Bbb H\Bbb…

In this paper we give an elementary proof of the local sum conjecture in two dimensions. In a remarkable paper [CMN, arXiv:1810.11340], this conjecture has been established in all dimensions using sophisticated, powerful techniques from a…

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\geq 3$ and $I$ an $\mathfrak{m}$-primary ideal of $R$. Let $r_J(I)$ be the reduction number of $I$ with respect to a minimal reduction $J$ of $I$. Suppose depth $G(I)\geq…

We generalize an improved Lech bound, due to Huneke, Smirnov, and Validashti, for the Buchsbaum-Rim multiplicity and mixed multiplicity. We reduce the problem to the graded case and then to the polynomial ring case. There we use complete…

In this note we sketch a proof of a fundamental conjecture, the codimension-three conjecture, for microdifferential holonomic systems with regular singularities. It states that any regular holonomic E-module extends beyond a…

Beginning from the resolution of the Dirichlet L function, using the inner product formula between two infinite-dimensional vectors in the complex space, the author proved the baffling problem--Hecke conjecture.

For positive integers $s$ and $L \geq 3$, Berkovich and Uncu (Ann. Comb. $23$ ($2019$) $263$--$284$) conjectured an inequality between the sizes of two closely related sets of partitions whose parts lie in the interval $\{s, \ldots, L+s\}$.…

Erd\H{o}s similarity conjecture was proposed by P. Erd\H{o}s in 1974. The conjecture remains open for exponentially decaying sequences as well as Cantor sets that have both Newhouse thickness and Hausdorff dimension zero. In this article,…

The direct summand conjecture asserts that if R is a regular local ring and S is a module-finite R-algebra containing R, then R is a direct summand of S as an R-module. It was previously known to be true if R contains a field or if dim R is…

We prove the following result, which is motivated by the recent work of Kurano and Roberts on Serre's positivity conjecture. Assume that (R,m) is a local ring with finitely-generated module M such that R/ann(M) is quasi-unmixed and contains…

We prove an inequality related to questions in Approximation Theory, Probability Theory, and to Irregularities of Distribution. Let $h_R$ denote an $L ^{\infty}$ normalized Haar function adapted to a dyadic rectangle $R\subset [0,1] ^{3}$.…

It is conjectured that the dual variety of every smooth nonlinear subvariety of dimension $> \frac{2N}{3}$ in projective $N$-space is a hypersurface, an expectation known as the duality defect conjecture. This would follow from the truth of…

If R is a local ring of dimension n, of a smooth complex variety, and if I is a zero dimensional ideal in R, then we prove that e(I)\geq n^n/lc(I)^n. Here e(I) is the Samuel multiplicity along I, and lc(I) is the log canonical threshold of…