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The secret lives of polynomial identities Bruce Reznick University of Illinois at Urbana-Champaign Beijing Forestry University May 26, 2016 Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities Mathematics is the art of logic and formulas are its poetry. I have always been fascinated by exact formulas, especially finite ones such as polynomial identities. They can seem easy and superficial, but the fact that they are true without hypotheses can make mathematicians uncomfortable. They don’t need us. They come into view unexpectedly, like meteorites on a vast Arctic plain. At first, they seem out of place, but after appropriate reflection, the best identities can signify deep and distant phenomena. “An idea which can be used only once is a trick. If you can use it more than once it becomes a method.” – George Po´lya and G´abor Szego¨ Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities I have always been fascinated by exact formulas, especially finite ones such as polynomial identities. They can seem easy and superficial, but the fact that they are true without hypotheses can make mathematicians uncomfortable. They don’t need us. They come into view unexpectedly, like meteorites on a vast Arctic plain. At first, they seem out of place, but after appropriate reflection, the best identities can signify deep and distant phenomena. “An idea which can be used only once is a trick. If you can use it more than once it becomes a method.” – George Po´lya and G´abor Szego¨ Mathematics is the art of logic and formulas are its poetry. Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities They can seem easy and superficial, but the fact that they are true without hypotheses can make mathematicians uncomfortable. They don’t need us. They come into view unexpectedly, like meteorites on a vast Arctic plain. At first, they seem out of place, but after appropriate reflection, the best identities can signify deep and distant phenomena. “An idea which can be used only once is a trick. If you can use it more than once it becomes a method.” – George Po´lya and G´abor Szego¨ Mathematics is the art of logic and formulas are its poetry. I have always been fascinated by exact formulas, especially finite ones such as polynomial identities. Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities They don’t need us. They come into view unexpectedly, like meteorites on a vast Arctic plain. At first, they seem out of place, but after appropriate reflection, the best identities can signify deep and distant phenomena. “An idea which can be used only once is a trick. If you can use it more than once it becomes a method.” – George Po´lya and G´abor Szego¨ Mathematics is the art of logic and formulas are its poetry. I have always been fascinated by exact formulas, especially finite ones such as polynomial identities. They can seem easy and superficial, but the fact that they are true without hypotheses can make mathematicians uncomfortable. Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities They come into view unexpectedly, like meteorites on a vast Arctic plain. At first, they seem out of place, but after appropriate reflection, the best identities can signify deep and distant phenomena. “An idea which can be used only once is a trick. If you can use it more than once it becomes a method.” – George Po´lya and G´abor Szego¨ Mathematics is the art of logic and formulas are its poetry. I have always been fascinated by exact formulas, especially finite ones such as polynomial identities. They can seem easy and superficial, but the fact that they are true without hypotheses can make mathematicians uncomfortable. They don’t need us. Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities At first, they seem out of place, but after appropriate reflection, the best identities can signify deep and distant phenomena. “An idea which can be used only once is a trick. If you can use it more than once it becomes a method.” – George Po´lya and G´abor Szego¨ Mathematics is the art of logic and formulas are its poetry. I have always been fascinated by exact formulas, especially finite ones such as polynomial identities. They can seem easy and superficial, but the fact that they are true without hypotheses can make mathematicians uncomfortable. They don’t need us. They come into view unexpectedly, like meteorites on a vast Arctic plain. Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities “An idea which can be used only once is a trick. If you can use it more than once it becomes a method.” – George Po´lya and G´abor Szego¨ Mathematics is the art of logic and formulas are its poetry. I have always been fascinated by exact formulas, especially finite ones such as polynomial identities. They can seem easy and superficial, but the fact that they are true without hypotheses can make mathematicians uncomfortable. They don’t need us. They come into view unexpectedly, like meteorites on a vast Arctic plain. At first, they seem out of place, but after appropriate reflection, the best identities can signify deep and distant phenomena. Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities It can be derived by using the commutative and associative law for complex numbers: ( )( ) (a + ib)(a − ib) (x + iy)(x − iy) ( )( ) (ax − by) + i(bx + ay) (ax − by) − i(bx + ay) ( )( ) = (a + ib)(x + iy) (a − ib)(x − iy) , And by setting (a, b) = (cos t, sin t), this identity shows that distance is invariant under a rotation of axes. Not bad for one identity. The two square identity is widely used in algebra and number theory: 2 2 2 2 2 2 (a + b )(x + y ) = (ax − by) + (bx + ay) Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities ( )( ) (ax − by) + i(bx + ay) (ax − by) − i(bx + ay) ( )( ) = (a + ib)(x + iy) (a − ib)(x − iy) , And by setting (a, b) = (cos t, sin t), this identity shows that distance is invariant under a rotation of axes. Not bad for one identity. The two square identity is widely used in algebra and number theory: 2 2 2 2 2 2 (a + b )(x + y ) = (ax − by) + (bx + ay) It can be derived by using the commutative and associative law for complex numbers: ( )( ) (a + ib)(a − ib) (x + iy)(x − iy) Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities

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The secret lives of polynomial identities Bruce Reznick University of Illinois at Urbana-Champaign Beijing Forestry University May 26, 2016 Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities Mathematics is the art of logic and formulas are its poetry. I have always been fascinated by exact formulas, especially finite ones such as polynomial identities. They can seem easy and superficial, but the fact that they are true without hypotheses can make mathematicians uncomfortable. They don’t need us. They come into view unexpectedly, like meteorites on a vast Arctic plain. At first, they seem out of place, but after appropriate reflection, the best identities can signify deep and distant phenomena. “An idea which can be used only once is a trick. If you can use it more than once it becomes a method.” – George Po´lya and G´abor Szego¨ Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities I have always been fascinated by exact formulas, especially finite ones such as polynomial identities. They can seem easy and superficial, but the fact that they are true without hypotheses can make mathematicians uncomfortable. They don’t need us. They come into view unexpectedly, like meteorites on a vast Arctic plain. At first, they seem out of place, but after appropriate reflection, the best identities can signify deep and distant phenomena. “An idea which can be used only once is a trick. If you can use it more than once it becomes a method.” – George Po´lya and G´abor Szego¨ Mathematics is the art of logic and formulas are its poetry. Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities They can seem easy and superficial, but the fact that they are true without hypotheses can make mathematicians uncomfortable. They don’t need us. They come into view unexpectedly, like meteorites on a vast Arctic plain. At first, they seem out of place, but after appropriate reflection, the best identities can signify deep and distant phenomena. “An idea which can be used only once is a trick. If you can use it more than once it becomes a method.” – George Po´lya and G´abor Szego¨ Mathematics is the art of logic and formulas are its poetry. I have always been fascinated by exact formulas, especially finite ones such as polynomial identities. Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities They don’t need us. They come into view unexpectedly, like meteorites on a vast Arctic plain. At first, they seem out of place, but after appropriate reflection, the best identities can signify deep and distant phenomena. “An idea which can be used only once is a trick. If you can use it more than once it becomes a method.” – George Po´lya and G´abor Szego¨ Mathematics is the art of logic and formulas are its poetry. I have always been fascinated by exact formulas, especially finite ones such as polynomial identities. They can seem easy and superficial, but the fact that they are true without hypotheses can make mathematicians uncomfortable. Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities They come into view unexpectedly, like meteorites on a vast Arctic plain. At first, they seem out of place, but after appropriate reflection, the best identities can signify deep and distant phenomena. “An idea which can be used only once is a trick. If you can use it more than once it becomes a method.” – George Po´lya and G´abor Szego¨ Mathematics is the art of logic and formulas are its poetry. I have always been fascinated by exact formulas, especially finite ones such as polynomial identities. They can seem easy and superficial, but the fact that they are true without hypotheses can make mathematicians uncomfortable. They don’t need us. Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities At first, they seem out of place, but after appropriate reflection, the best identities can signify deep and distant phenomena. “An idea which can be used only once is a trick. If you can use it more than once it becomes a method.” – George Po´lya and G´abor Szego¨ Mathematics is the art of logic and formulas are its poetry. I have always been fascinated by exact formulas, especially finite ones such as polynomial identities. They can seem easy and superficial, but the fact that they are true without hypotheses can make mathematicians uncomfortable. They don’t need us. They come into view unexpectedly, like meteorites on a vast Arctic plain. Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities “An idea which can be used only once is a trick. If you can use it more than once it becomes a method.” – George Po´lya and G´abor Szego¨ Mathematics is the art of logic and formulas are its poetry. I have always been fascinated by exact formulas, especially finite ones such as polynomial identities. They can seem easy and superficial, but the fact that they are true without hypotheses can make mathematicians uncomfortable. They don’t need us. They come into view unexpectedly, like meteorites on a vast Arctic plain. At first, they seem out of place, but after appropriate reflection, the best identities can signify deep and distant phenomena. Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities It can be derived by using the commutative and associative law for complex numbers: ( )( ) (a + ib)(a − ib) (x + iy)(x − iy) ( )( ) (ax − by) + i(bx + ay) (ax − by) − i(bx + ay) ( )( ) = (a + ib)(x + iy) (a − ib)(x − iy) , And by setting (a, b) = (cos t, sin t), this identity shows that distance is invariant under a rotation of axes. Not bad for one identity. The two square identity is widely used in algebra and number theory: 2 2 2 2 2 2 (a + b )(x + y ) = (ax − by) + (bx + ay) Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities ( )( ) (ax − by) + i(bx + ay) (ax − by) − i(bx + ay) ( )( ) = (a + ib)(x + iy) (a − ib)(x − iy) , And by setting (a, b) = (cos t, sin t), this identity shows that distance is invariant under a rotation of axes. Not bad for one identity. The two square identity is widely used in algebra and number theory: 2 2 2 2 2 2 (a + b )(x + y ) = (ax − by) + (bx + ay) It can be derived by using the commutative and associative law for complex numbers: ( )( ) (a + ib)(a − ib) (x + iy)(x − iy) Bruce Reznick University of Illinois at Urbana-Champaign The secret lives of polynomial identities

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