Lihchung Wang - Academia.edu (original) (raw)

Papers by Lihchung Wang

Research paper thumbnail of Divisors of Generic Hypersurfaces of General Type

Taiwanese Journal of Mathematics

ABSTRACT We prove that a generic hypersurface of general type in P n for 3 does not contain a red... more ABSTRACT We prove that a generic hypersurface of general type in P n for 3 does not contain a reduced irreducible divisor which admits a desingularization having nef anticanonical bundle.

Research paper thumbnail of On the Jacobian Conjecture

Taiwanese Journal of Mathematics

Let k be an algebraically closed field, and let f: k"-+ k" be a polynomial map. Then f is given b... more Let k be an algebraically closed field, and let f: k"-+ k" be a polynomial map. Then f is given by coordinate functions fl, f,, where each fi is a polynomial in n variables X1, X,. Iffhas a polynomial inverse g (g 1, ,), then the determinant of the Jacobian matrix f/OXj is a non-zero constant. This follows from the chain rule: Since f 0 is the identity, we have X 0i, f,), so X t=l This says that the product tXs is the identity matrix. Thus, the Jacobian determinant off is a non-vanishing polynomial, hence a constant. The Jacobian conjecture states, conversely, that if the characteristic of k is zero, and if f= (f,..., f,) is a polynomial map such that the Jacobian determinant is a non-zero constant, then f has a polynomial inverse. The problem first appeared in the literature (to my knowledge) in 1939 in [11] for k C. Many erroneous proofs have emerged, several of which have been published, all for k C, n 2. The conjecture is trivially true for n 1. For n > 1, the question is open. There has been a vigorous attempt by S. Abhyankar and T.-T. Moh to solve the problem for n 2. In this case it is known that the Jacobian conjecture is equivalent to the assertion that whenever f (f, f2) satisfies the Jacobian hypothesis, the total degree off divides that off2, or vice versa. Abhyankar and Moh have obtained a number of partial results by looking at the intersection of the curves fl and f2 at infinite in p2. Moh has proved, in fact, that the conjecture is true provided the degrees of A and A do not exceed 1 [15].

Research paper thumbnail of SQUARE-FREE Qk COMPONENTS IN TTM

Taiwanese Journal of Mathematics

Research paper thumbnail of Tractable rational map public-key system

Research paper thumbnail of A remark on divisors of Calabi–Yau hypersurfaces

Asian Journal of Mathematics, 2000

We prove that a non-singular hypersurface of degree > n + 1 in P n for n > 4 does not contain a r... more We prove that a non-singular hypersurface of degree > n + 1 in P n for n > 4 does not contain a reduced irreducible divisor which admits a desingularization having nef anticanonical bundle.

Research paper thumbnail of A Simple Proof of Thue's Theorem on Circle Packing

A simple proof of Thue theorem on Circle Packing is given. The proof is only based on density ana... more A simple proof of Thue theorem on Circle Packing is given. The proof is only based on density analysis of Delaunay triangulation for the set of points that are centers of circles in a saturated circle configuration.

Research paper thumbnail of On group structure associated to Jacobian pairs with mixed leading forms

Journal of Pure and Applied Algebra, 1997

... Denote by Jx.v(f,S) ths Jacobian determinant of and g. The wellknown Jacobian conjecture stat... more ... Denote by Jx.v(f,S) ths Jacobian determinant of and g. The wellknown Jacobian conjecture states "If Jx,vU ij) is a nonzero constant, then C[x,] = C[f,g ". For a survey of the Jacobian problem, we will refer the readers to [3]. One of the most effective approaches for this problem is ...

Research paper thumbnail of A fast mental poker protocol

Journal of Mathematical Cryptology, 2000

We present a fast and secure mental poker protocol. It is twice as fast as similar protocols, nam... more We present a fast and secure mental poker protocol. It is twice as fast as similar protocols, namely Barnett-Smart's and Castellà-Roca's protocols. This protocol is provably secure under DDH assumption.

Research paper thumbnail of Divisors of Generic Hypersurfaces of General Type

Taiwanese Journal of Mathematics

ABSTRACT We prove that a generic hypersurface of general type in P n for 3 does not contain a red... more ABSTRACT We prove that a generic hypersurface of general type in P n for 3 does not contain a reduced irreducible divisor which admits a desingularization having nef anticanonical bundle.

Research paper thumbnail of On the Jacobian Conjecture

Taiwanese Journal of Mathematics

Let k be an algebraically closed field, and let f: k"-+ k" be a polynomial map. Then f is given b... more Let k be an algebraically closed field, and let f: k"-+ k" be a polynomial map. Then f is given by coordinate functions fl, f,, where each fi is a polynomial in n variables X1, X,. Iffhas a polynomial inverse g (g 1, ,), then the determinant of the Jacobian matrix f/OXj is a non-zero constant. This follows from the chain rule: Since f 0 is the identity, we have X 0i, f,), so X t=l This says that the product tXs is the identity matrix. Thus, the Jacobian determinant off is a non-vanishing polynomial, hence a constant. The Jacobian conjecture states, conversely, that if the characteristic of k is zero, and if f= (f,..., f,) is a polynomial map such that the Jacobian determinant is a non-zero constant, then f has a polynomial inverse. The problem first appeared in the literature (to my knowledge) in 1939 in [11] for k C. Many erroneous proofs have emerged, several of which have been published, all for k C, n 2. The conjecture is trivially true for n 1. For n > 1, the question is open. There has been a vigorous attempt by S. Abhyankar and T.-T. Moh to solve the problem for n 2. In this case it is known that the Jacobian conjecture is equivalent to the assertion that whenever f (f, f2) satisfies the Jacobian hypothesis, the total degree off divides that off2, or vice versa. Abhyankar and Moh have obtained a number of partial results by looking at the intersection of the curves fl and f2 at infinite in p2. Moh has proved, in fact, that the conjecture is true provided the degrees of A and A do not exceed 1 [15].

Research paper thumbnail of SQUARE-FREE Qk COMPONENTS IN TTM

Taiwanese Journal of Mathematics

Research paper thumbnail of Tractable rational map public-key system

Research paper thumbnail of A remark on divisors of Calabi–Yau hypersurfaces

Asian Journal of Mathematics, 2000

We prove that a non-singular hypersurface of degree > n + 1 in P n for n > 4 does not contain a r... more We prove that a non-singular hypersurface of degree > n + 1 in P n for n > 4 does not contain a reduced irreducible divisor which admits a desingularization having nef anticanonical bundle.

Research paper thumbnail of A Simple Proof of Thue's Theorem on Circle Packing

A simple proof of Thue theorem on Circle Packing is given. The proof is only based on density ana... more A simple proof of Thue theorem on Circle Packing is given. The proof is only based on density analysis of Delaunay triangulation for the set of points that are centers of circles in a saturated circle configuration.

Research paper thumbnail of On group structure associated to Jacobian pairs with mixed leading forms

Journal of Pure and Applied Algebra, 1997

... Denote by Jx.v(f,S) ths Jacobian determinant of and g. The wellknown Jacobian conjecture stat... more ... Denote by Jx.v(f,S) ths Jacobian determinant of and g. The wellknown Jacobian conjecture states "If Jx,vU ij) is a nonzero constant, then C[x,] = C[f,g ". For a survey of the Jacobian problem, we will refer the readers to [3]. One of the most effective approaches for this problem is ...

Research paper thumbnail of A fast mental poker protocol

Journal of Mathematical Cryptology, 2000

We present a fast and secure mental poker protocol. It is twice as fast as similar protocols, nam... more We present a fast and secure mental poker protocol. It is twice as fast as similar protocols, namely Barnett-Smart's and Castellà-Roca's protocols. This protocol is provably secure under DDH assumption.