ASU SoMSS

Interests

My research interests include number theory, algebraic geometry, especially Diophantine problems and Galois theory. I am also interested in algebraic topology, medicine, brain study, cryptography, and AI.

Papers

1, Bremner, A; Xuan, T. N. The equation (w + x + y + z)(1/w + 1/x + 1/y + 1/z) = n.
International Journal of Number Theory, Vol. 14(2018), No. 05, 1-18. [PDF]
In this paper, we gave a negative answer to a conjecture of Bremner, Guy, and Nowakowski. This is a nontrivial application of p-adic analysis and elliptic curve theory.
It is based on an ideal of Michael Stoll.

Note: Professor Andrew Bremner computed all the solutions of (w+x+y+z+w)(1/w+1/x+1/y+1/z)=4m^2 with 4|m-2 and 4m^2<20000 except 10000 and 15376.

Problem 1: Find a positive integer solution of (w+x+y+z)(1/w+1/x+1/y+1/z)=n with n=10000 or 15376

Question 1: For n>16 and n is not of the form 4m^2 or 4m^2+4 where m is not 2 mod 4 then the equation
(w+x+y+z)(1/w+1/x+1/y+1/z)=n always has solutions in positive integers.

Question 2: For n>25 then the equation (x+y+z+w+e)(1/x+1/y+1/z+1/w+1/e)=n always has solutions in positive integers.

2, Bremner, A; Xuan, T.N. An interesting quartic surface, everywhere locally solvable, with cubic point but no global point.
Publ. Math. Debrecen, Vol. 93/1-2(2018). 253-260. [PDF]

In this paper, we studied the equation x^4+7y^4=14z^4+18w^4 which is the first example in the family of diagonal quartic surfaces which has nontrivial solutions in a cubic number field or Q_p for all prime p, but has no nontrivial solutions in rational numbers. Other known examples by Swinnerton-Dyer and Martin Bright do not seem (unproved) to have these properties.

Note: Professor Andrew Bremner found the surface x^4+7y^4=14z^4+18w^4 by searching for the family ax^4+by^4=cz^4+dw^4 where abcd=square, but only one example is found.

Question 3: Is there an infinitely family of quartic surfaces of the form ax^4+by^4=cy^4+dz^4 satisfy the following properties:
i, unsolvable in rational numbers except xyzw=0
ii, solvable in Q_p for every prime p
iii, solvable in a cubic number field.

3, Phd Thesis. [PDF]

Preprints

1, With A. Bremner, The equation y^2=x^6+k with k=-39 or k=-47. [PDF] (16 pages)
In this paper, we solved the two open cases of the equation y^2=x^6+k where k in range |k|<51. All the other values of k within the range [-50,50] were solved by Bremner and Tzanakis . This is an application of computational algebraic number theory and the elliptic curve Chabauty method.

Note: Bremner and Tzanakis found 20 rational points on y^2=x^6+1025 which are (+/-x,+/-y)= (2,33), (1/4,2049/64), (5/2,285/8), (8,513), (20/91,24126045/913).
The curves y^2=x^6+1025 seems to have the maximum number of points in the family y^2=x^6+k where |k|<250000.

Problem 2: Show that the only rational solutions to y^2=x^6+1025 are (+/-x,+/-y)= (2,33), (1/4,2049/64), (5/2,285/8), (8,513), (20/91,24126045/913)
Some other problems (rational points on curve of genus > 1):

i, Fermat curve x^4+y^4=dz^4 where d= 82, 97 or 257.
The equation x^4+y^4=dz^4 defines a curve of genus 3. In general, it is hard to find all rational solutions.
x^4+y^4=17z^4 is a problem in Serre's Lectures on Mordell-Weil Theorem and was solved by Flynn. In the same paper, he also suggested a general approach to attack

x^4+y^4=dz^4. Using Magma, it's possible to deal with the case d=97 but d=257 is still open.

ii, Rational points on curve y^2-y=x^5-x.
All integral points on the curve y^2-y=x^5-x are found by Bugeaud and others, but finding all rational points is open.

iii, Rational points on the curve y^2=2x^6+4x^5+36x^4+16x^3-45x^2+190x+1241.
The curve y^2=2x^6+4x^5+36x^4+16x^3-45x^2+190x+1241 paramatrizes the all trinomials ax^8+bx+c over a field of character 0 with
the Galois group G_{1334}. See Elkies.

2, With A. Bremner, The equation x/y+py/z+z/w+pw/x=8np. [PDF] (9 pages)

This paper is another example of the p-adic method which was used to study the equation (x+y+z+w)(1/x+1/y+1/z+1/w)=n. Currently, to our knowledge there are only 2 known homogeneous rational forms, (w+x+y+z)(1/w+1/x+1/y+1/z) and x/y+py/z+z/w+pw/x, which are shown to have no rational solutions in positive integers.

3, With A. Bremner, Cubic points on quartic curves. [PDF] (14 pages)

In this paper, we studied the equation F(x^2,y^2,z^2)=0 in cubic number fields where F(x,y,z) is a homogeneous polynomial of degree 2 with rational coefficients.
The equation F(x^2,y^2,z^2)=0 defines a curve of genus 3. A necessary condition is given when the equation F(x^2,y^2,z^2)=0 has a solution in a cubic number field. This extends some old results by Cassels and Bremner.

Note: Using an algorithm by Cassels and Bremner, we computed solutions to the solution of x^4+y^4=Dz^4 for many values of D up to 10000. The strategy here is to search for a cubic field Q(t) where at^3+bt^2+ct+d=0 such that X^4+Y^4=D has a solution in Q(t). This was accomplished by searching for rational points on some 64 degree homogeneous polynomials in a,b,c,d.
Example:
The equation x^4+y^4= 9281z^4 has a solution
x=(-5654t^2 + 14435t - 1735)/29
y=(-92t^2 +t - 30)/23
z=t^2+1
where -123119/4682t^3+158196/2341t^2-74015/4682t+1=0
All computation turns out that when the rank of the curve X^4+Y^2=D is at least 2, there is always a solution in a cubic number field. But this seems
every hard to prove.

Question 3: Is there a value of D such that the rank of the curve X^4+Y^2=D is at least 2 but the equation x^4+y^4=Dz^z^4 has no
solutions in any cubic number field.

Question 4: Is there a value of D such that the curve x^4+y^4=Dz^4 has a solution in Q_p for every p, a solution in cubic number fields but has no solutions
in rational number.
Originally, Professor Andrew Bremner showed that D = 4481 and D = 5617 have the properties in Question 3, but 30 years later, he found a mistake in his computation.

Problem Solving

These are some notes on math problem solving I do in free time. It's a collection of fun math problems. I created some of them while others are from various sources like Putnam , USAMO, IMO ,VMO:
-Functional Equations [PDF]

-Polynomials [PDF]

Also:

Putnam's problems and solutions by Kiran S. Kedlaya
IMO collection of problems and solutions from 1959-2004
Collection of geometry competition problems
My favorite book on inequalities
Hard Problems: The Road to the World's Toughest Math Contest