Bound A Sum Of Squares - Not As Easy As You Think

Algebra Level 5

Consider polynomials of the form $x^{2014} + a_{2013} x^{2013} + \ldots + a_1x + 1$ , with real coefficients and having at least one real zero. Determine the smallest possible value of $\sum_{i=1}^{2013} a_i^2$ .

Round your answer to 3 decimal places.

The answer is 0.002.

2 solutions

Aviroop Pal
Jan 26, 2015

Clearly suppose 1 is the only real root of the polynomial. We get a1+a2…+a2013=-2. Now apply cauchy schwarz inequality.

How do you know that 1 must be the real root of the polynomial?

Why can't another root give us a smaller sum of squares?

Calvin Lin Staff - 6 years, 3 months ago

suppose we consider the case when the degree of the polynomial is 4. we first solve the following problem. the polynomial f(x)=x^4+ax^3+bx^2+cx+1 with real coefficients has atleast one real root. we want to find the minimum value of a^2+b^2+c^2 . suppose f(x) has one real root the f(x)=(x+r)(x^3+px^2+qx+1/r) where r,p,q are real numbers and r is not equal to 0. now (x+r)(x^3+px^2+qx+1/r)=x^4+(r+p)x^3+(rp+q)x^2+(qr+1/r)x+1 we have to minimise (r+p)^2+(rp+q)^2+(qr+1/r)^2 constraints are r is not equal to zero. we can do this for any even degree polynomial. in this problem the degree of the polynomial is 2014. minimising this expression needs lot of hard work. this method surely works. may not be the best way of doing things.

Srikanth Tupurani - 10 months, 3 weeks ago

Adit Mohan
May 11, 2014

obviously the sum will be minimum when all the terms are equal or magnitude of the coefficients are equal.
i worked backward from here and noticed that.
x=-1 and $a_{even}$ =-2/2013 and $a_{odd}$ =2/2013 worked beautifully.
sum = 2013(2.2/2013.2013) = 4/2013 = 0.00198

It is not obvious that the sum will be minimum when all the terms are equal in magnitude.

Calvin Lin Staff - 6 years, 3 months ago

Bound A Sum Of Squares - Not As Easy As You Think

The answer is 0.002.

2 solutions

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